"Is my estimation correct, or are there any pitfalls I haven't yet realized which would make this quick-n'-dirty path prohibitive?"

I do not think that plan will save you any time getting cool.
It will take you all that time to do the proper things to your system to put the R12 into it... if you stick the other in you will need to start over on many things including buying a new Rec/Dryer, all the orings and install them, flush again,... better to spend your spare time learning what the physical constraints are concerning the physics of AC and our particular Delco compressor ( no oil sump in it.. so oil miscibility is Critical for lubrication.
Start by reading all you can find at Aircondition.com knowledge base..
Start with the concept of leak testing with nitrogen and R22...
Then we can point out good threads on this site to read.

I also suggest right out of the shut you go to the epa site and get your open book test and license to buy R12... about $20 .. lots of theads about that here also...
and it spells out some of the best reasons not to use blends like Freeze 12... which is misnamed in order to mislead you .. it is 80 percent 134a...