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By digging into the code and going a little lower-level I was able to do effectively the same thing while close-enough to the lower-level code that it accepted the maxterms argument (of course, you shouldn't have to do this at all and I would consider it a bug):

sage: from mpmath import hyper
sage: from sage.libs.mpmath import utils as mpmath_utils
sage: mpmath_utils.call(hyper, [4.14 + 15*I, -3.14 + 15*I],[1. - 1.12e7*I], -500000, maxterms=10^6)
-608213.273405350 - 24201.2928264098*I

I make no guarantee as to the accuracy of the result or if it's even expected be convergent for the given $a$'s and $b$'s.