On our 3456a's that get adjusted now and then, there is never a problem doing the adjustment with the cal pots. Just bring the pot to where about you want it, then swing the wiper back and forth the area where you want it...I'll give it a few "cleaning wipes" if you see any jitters - you'll know when you've got the new wiper contact area cleaned up if it hasn't been adjusted in ages.
Then give it the final nudge trim, remove the tool and stand back, wait a minute or two and it should be OK. You might not touch that pot again for a years or decades. They work fine for decades IF they are higher quality pots - forget the cheap plastic pots.
DigilentMinds: There is a small dust cover on the front of a 3456a already for the cal pots access - no need to remove the covers. The meter doesn't really care if that cover is in place or not during cal. So you don't really have to remove any of the main instrument covers to do a cal. The pots are positioned that way on purpose.
We build trimmers into some Vref also and I can vouch for the Mr. Pettis' recommendation on the Bourns PWW pots - these are excellent pots, run around $12 or $15 each and give you an excellent method of trimming a circuit without adding digital noise. They also come with a natural non-volatile memory and are much, much quieter than any digital-tuned, diffused resistor array in a DAC (for instance) - AND pots don't contaminate a pure analog system with digital hash.
Bourns has some excellent literature on how to design a trimmer pot circuit correctly - a good read if you're not familiar with the procedure - the common mistakes we see with young players is not considering what happens when the wiper is adjusted into the max / min stops, and building a circuit with a "dry" wiper contact (not enough current flow - usually you want around 20~40uA minimum current on the wiper):
https://www.bourns.com/pdfs/onlinepotentiometerhandbook.pdfWatch out on Vishay pot specs also - First they will claim some low TC for the pot, and then realize the TC of the wiper will be 50 or 100ppm.
HINT: Design your trimmer circuit correctly and the trim pot TC won't have a huge effect... Lots of older equipment works this way and will often beat the newer stuff on stability, noise and TC.