Yes,you are correct,but note that he already has an overheating problem.Depends entirely on current being drawn from the second tranny's secondary winding.
I have seen "back to back" transformers used on many occasions,but unmatched windings are still a recipe for disaster.
There is no way of knowing what the current ratings of the low voltage windings are on his "junk box" transformers.
There is also the matter of the winding with 300v on it being run at above its voltage ratings.
Probably not a problem,but not to be encouraged.
The OP probably has a fair chance of picking up a secondhand transformer on eBay,with all the correct windings.