Worse still, practically all such “codes” use shortcuts, simplifications, or outright non-physical spacetimes to reduce the computer power required.
You and I are looking for the same thing, so if you do find a good reference please reply!
The sum is intractable not because it's big but because it can't be sanely approximated by lesser sums, due to the nature of what you're summing - an exponent of i * the integral of the Lagrangian over each spacetime field configuration. This vary likes crazy so if you skip the some parts you might get a completely different result.
The trick to anyhow get anything out of this is to restrict yourself to observables that can be calculated from the ground state, which turn out to be accessible by doing the field integrals over imaginary time instead. This is a trick of calculus, it's not really "non-physical", although it is a pretty crazy method :) This has allowed practitioners to calculate hadron masses for example, it's getting better and better over the decades.