Electrical Help Please

I'm no good with electronics so I need some help with my '68 Dart with a 505"/440 stuffed into the engine bay with the battery moved to the trunk. First time I drove it I stopped to put gas in the car. Then I went to show the car to my father-in-law. Then I stopped at Wendy's on the way home. The battery died at Wendy's and I had to call my wife to jump me off.

The car is slow to start like the timing is too high or like it has a lot of compression. It eventually kills the battery. I'm afraid to drive the car because I don't want to get stranded. I have hooked a Battery Tender to the battery so it will give a slow trickle charge and the battery will be fresh every time I drive it. I also moved the timing temporarily so it's easier to crank. It still seems to run good but it's been 20 degrees outside so I haven't really done much to it yet like actually checking the timing with a timing light.

I called the previous owner and he said the same thing had been happening to him. He suggested a voltage regulator. Would a bad voltage regulator cause the battery not to hold a charge? I think that's a stock type MOPAR voltage regulator on the firewall in the pic. Seems like the alternator might be bad? How can I tell. Someone mentioned to remove the positive battery cable from the battery while it was running? What would that show?

Thanks for the help in advance.

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