I've done this on three trip meters with good results:

With the speedometer out of the cluster, find the bent flat-steel spring that the reset "button" ultimately bears against. Without removing the spring from it's home, flatten it's angle bend with needle nose pliers. Flatten it partially or fully; your choice. This removes the strong "click" resistance you feel when resetting the trip meter. The stress of overcoming that resistance is what I believe breaks the trip reset "button".

Details:
Reset "button" shaft bears against steel pin (sideways, not end-wise). Pin has small coil spring on each end that provides basic resistance, no worries on those. Pin also bears against flat spring, its job is to provide the click feeling when resetting. That's the nasty bugger that has to be weakened.

I've done this on three trip meters while replacing the broken reset button. Reset button has not failed on either car in spite of the fact that on two of those cars I use/used the reset 3-5 times weekly for years.
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Sven: '89 245 NA, 951 ECU, expanded air dam, forward belly pan reaches oem belly pan, airbox heater upgraded, E-fan, 205/65-15 at 50 psi, IPD sways, no a/c-p/s belt, E-Codes, amber front corner reflectors, aero front face, quad horns, tach, small clock.