Hard downshift from 2nd gear to first. A-833 4-speed.

Ok so with the car moving.
I assume you know how synchronizers work, but humor me.
The two FOUR working parts of the Synchronizers are;
the Brass rings
the Brake cones
the three struts, and
the Energizer springs.
If any of those parts fail to do their jobs, You have problems, and a couple of different ones at that.
In your case, you do not mention any gear-clash or grinding noises.
Now, consider this:
When the car is stopped and your foot is OFF the clutch, and the engine is idling:
1) The driveshaft is Not turning so neither is the output shaft, nor the two synchronizer assemblies that are mounted on it.
2) the engine is at around 750 rpm, and so are the clutch and, the trans input gear. The cluster and all three gears are spinning at various speeds.
If you tried to engage a gear, by NOT first clutching it, the brass ring would have to try to slow the engine down, to zero rpm so that the big slider could engage the clutch teeth of whatever gear you were trying to engage. Obviously the relatively small brass-ring could not do that, so some smart guy invented the De-clutching mechanism.
>>With the Clutch pedal down, and sufficient clutch departure, the input gear is allowed to be slowed to a crawl by the thick gear-oil until the brass ring is able to grab the brake cone, synchronize the gear to it's speed, and BadaBoom the slider slips over.

Every upshift gear change works the same. Except, the designer of the trans expects the driver to learn to to keep the rpm difference in a realistic operating window. For instance, if a particular gear is synchronized at 30 mph @3000 rpm, then there's no good reason to try and select that gear with the tach at 6000. Any little friction in the clutch will make impossible.
So then, while you are accelerating thru the gears, the driver is expected to allow the rpm to come down, into the window. If you miss it and the rpm drops too low, again, any clutch drag is gonna make it hard for the brass to synchronize.
Now, for downshifts, the Brass ring works in the same way, But in this case, the operator is expected to bring the rpm UP, into the window of synchronization, to give the brass a chance to do it's thing.
Most of us just blip the throttle and hesitate the shift, waiting for synchronization to happen. If you miss the window, you just bring the rpm back up.
So, what I'm saying is that a great deal of shifting is on the operator. Eventually, this becomes automatic.
So far, I have talked about technique and clutch departure.

Now we'll talk about the parts.
First how it's supposed to work.
Synchronization is 100% dependent on the brass, which by friction, lock onto the brake cones. So the first order is to squeeze the oil out. Synthetic oil is very difficult to squeeze out, and even when it's out, it leaves a micro-film of slippery molecules stuck on the brake cones that is as good as impossible to remove. Even when you try to wash it out with solvent. That is some slipperychit. But say you got it all out, and the brass has squeezed out whatever oil you installed. Now the brass is in intimate contact with the brake; and it's inner dimensions exactly match the brake cones, and they are not egg-shaped nor out of round. So then, by friction, they bring the rotational speed of the gear attached to the brake, up or down, to match the speed of the brass. As soon as it is within a few rpm, the slider jumps forward to engage the clutch teeth, and synchronization has occurred.
Ok now, what pushes the brass ahead onto the brake?
That would be the struts. but there are two other parts involved in this process, namely, the matching notches in the slider and the energizer springs.
The springs keep the struts jammed into the slider notches making it act as one unit, spinning at driveshaft speed. The faster the driveshaft spins, the harder the struts are flung into the notches, but in the grand scheme of things, this is not so big a deal, BUT, the springs gotta be doing their jobs, cuz if they are weak, the slider will move ahead too soon, causing the teeth to butt and grinding is the result.

Now, you did not mention grinding. So then everything seems to be working as designed and the only hick-ups can be lack of friction, or technique.
As to friction, When I rebuild one of these, I chuck the gears in a lathe and deglaze the cones with 100>120 emery cloth, just a couple of swipes. and I bias the swipes so that the brass rings screw onto the brakes in the upshift direction. and I make sure the brass works by dry-fitting them onto their intended brakes, always giving special attention and priority to Second gear. Cuz I know,
Second is the single-most-used gear in that box.
Fourth is only used for cruising, and
Third is mostly just used to get to fourth.
and First is mostly just downshifted into, so I bias that brake for down-shifting. When I get done, my brakes shift like lightning for a long time. But, I have been driving the A833 almost continuously since 1970, so, I got pretty good technique, lol.
And finally is oil.
I find that All my A833's have loved Dextron II.
and all have tolerated 85/90 EP oil. But none have liked 140 EP, and the one I currently drive, absolutely hates full-synthetics.
With Dextron,
you gotta have adequate clutch-disc departure, cuz you'll get no help from it, to slow the gears. Whereas;
the 140 will slow the gears real fast, and it's real easy to miss the synchronization window. and
85/90 works pretty good.
So my recipe is half Dextron/half 85-90, cuz the EP oil slows wear on the cluster pin.

Now in your case, down-shifting into First from Second with the car moving;
This is likely the most difficult shift to make work right.
Here's what all has to happen;
1) The Rpm has to be increased to get into the synchronization window
2) the clutch departure has to be adequate, so that the disc does not drag.
3) the engine rpm must NOT come down too fast after the blip
4) the trans-oil has got to get out of the way, and
5) the brass has got to bite on the brake, and
6) the struts have to not want to come out of the notches until it is time.
7) and the wild card is that the 3-4 internal shift lever on the older covers, has to properly detent so that the interlock-pin is not being tickled. This is what @dadsbee was mentioning in post #2.

That's a lottachit going on right there.
Since you say that you have a Brewers Transmission, Ima thinking he has done his part in making that unit work. He knows his stuff.
So then, if I had to guess;

I am confident that the problem is Not #6, and
reasonably confident Not #3 nor #7.
Had you never installed synthetic, I would have put #5 with #3 as reasonably confident, but like I said earlier, I had to disassemble my trans to wash that slipperychit out of my box.

Therefore, I'm leaning towards numbers 1, and/or, 4>5, in that order; with numbers 2 and 7 as possibilities.

Happy HotRodding

I almost forgot
The notches in the sliders, and the struts are a matched set, AND the old style is different from the new. The old style struts have a large drive hump on them, and a matching deep notch. Whereas the notches in the new-style slider is pretty shallow. Otherwise, those sliders are physically interchangeable, while the struts go with the hubs. IDK what would happen with old style struts in new style sliders. However Your symptoms do not point to a mix up, so I wouldn't think about it. I just thought you might like to know, lol.