That ups the limit to 596 polynomials x 6 coefficients ~= 3500 coefficients total, per method. Tested working with five sets of 596 polynomials ~= 17000 coefficients total. So it can be done, but might need some shoehorning.
That might be enough of a reason to flip to the external file solution.
The byte-coefficients-in-one-big-string is still a plausible option. If the string did ever grow > 64 kB (ie fourfold) then it could be split amongst methods and/or code files.
It'd be useful to have a more detailed description than just "15k integers", like how large are the polynomials, does that size vary, how many are there, what are these subsets, etc. But presumably there must be limits to what you can reveal, so... no worries, we'll forge on in darkness