I'm not an expert on modular forms, but maybe I can help point you in the right direction. If you take a look at the source code for hecke_operator_on_basis, you will see that it just computes some auxiliary objects and then calls _hecke_operator_on_basis, defined in that same source file. Note that hecke_operator_on_basis uses the first element of the list to compute the auxiliary objects -- that seems to be the root cause of the asymmetry.
That function, in turn, calls hecke_operator_on_qexp on each of the elements of the list. The error message says things are breaking on line 105/106, and with a little testing I determined that the break occurs when hecke_operator_on_qexp is applied to your h (with g being the first element in the list).
This is where my ability to follow starts breaking down, but that line is using a number of coefficients of the series (f there), and it seems likely that the error is occurring because the precision of your g is much larger than that of h. So would you say this is a bug? If so, you should report it!