The best way to trim a wheatstone bridge digitally?
First of all you should really get better resistors. 5% with a strain gauge is just ridiculous.

To "trim and calibrate" your measurement you should just leave your gauges be, i.e. leaving them in a known state, measure the output and save it in a variable in your micro. This value should then be subtracted from all subsequent measurements. If you can you should measure the offset periodically, but if you want to fight just that 5% and nothing else a once in a lifetime measurement should do.

Please note that measuring Vr1r2 with respect to ground might appear appealing since you don't need to know the gauges state, but hey you are using a Wheatsone bridge to contain things like temperature drift and whatever, measuring your offset with respect to ground just defeats the bridge purpose.

Really, get better resistors.