You don't specify "logical budget". If you want some spot checks, I'd suggest looking into the
DMMCheck Plus. If you want to test the full range, then I'd look for a used
Fluke 5100/5101 A/B. If you're patient, you might be able to get it for something like €1k plus a few hundred € shipping. It's huge, heavy and might need some repairs given its age, though. I'm not aware of a cheaper option that can test all ranges of a multimeter. It can be operated via GPIB if you get one with the right option. If you want to buy the same new, consider a second mortgage on your house.
Otherwise I'd use a programmable power supply for DCV and DCI, a function generator for ACV and ACI (limited current range), and make a board with a bunch of decent resistors and banana jacks to test resistance. Something like the
Toellner device offered here might also work for low frequency ACI. But testing 5.5 digit DMMs this way is a bit of a challenge because these sources may not be stable enough. I have done it, and what I had to do was to hook up two meters in parallel / series, record over a long time, and then calculate correlation coefficient between the two meters. Also, it limits voltage ranges for both DCV and ACV to about 30V DC and about 7 Vrms AC. For DCV you can put multiple channels / power supplies in series, and higher voltage power supplies aren't so hard to come by. But I haven't found an easy way for generating ACV up to say 750 Vrms 1 kHz outside dedicated calibrators.
For consistency I would try to automate as much of the test as possible. Hence my suggesting programmable equipment. Just have the automation prompt for the measured value if the device under test does not support remote control (like most handhelds).