It it too much to ask for a complete and clean installed takeover solution?
Looking for a power option with a battery backup that will run two keypads (Main and a TS1) and one (maybe two) takeover module(s) from a central location. Would love it to be economical (under $50) and to be in a wall-mounted can if possible.
What options do you suggest? Any solutions from Elk or Altronix specifically?
The difficulty is not the takeover modules. A cheap Honeywell HPL624 can power your takeover modules and handle the battery back-up with a built in charging circuit.
The issue is that the panel and TS1 run off a non-standard voltage transformer. While they may run off of 12 VDC, you would be voiding the warranty and asking for power problems. So I cannot recommend a similar solution.
If you want to get rid of the old control panel though and clean up the space a bit, the small HPL624 (or HPS3 depending on current requirements) is a great option.
I see, because the panels run at 14Vish, then I would need a power board capable of supplying that rather than 12V? Like a switching power supply? That’s annoying. 
I’ve tried running it off my old system board (12V) and am definitely running into power issues running the second panel.
Another question though. The wire I have running between the central control and the panel locations is something like 18/4 AWG. On red/black I get 12-13 with no load. On the other two, I get another 5V each. Any chance that adding one of those into the positive lead on the panel increases the voltage enough to power it properly?
Doubling conductors will reduce voltage drop by a good deal, yes.
You can get a good indication of performance by using a tool like this one. You would need an accurate idea of cable length.