There are different types of rechargeble 9 V batteries. Some have 7 cells and some have 6 cells and thus a slightly lower voltage. It than also depends on the meter at which voltage they complain about insufficient voltage and show the battery symbol. It may happen that a meter will show low battery with a 6 cell block rather fast, as the nominal voltage of NiMH of some 1.2 to 1.24 V is about there where alkaline cells are considered empty. If the DMM is OK with some 7 V, than things are OK with a rechargible battery. Ideally this should be a low self discharge one, like the pre-charged ones, not an more old style high current one. In most cases there is not that much money to be saved, as a DMM usually does not need that many batteries. It still depends on the meter and use. The rechargible cells also have a limited life time and may thus only get very few charge cycles. The other reason to use them is the lower chance of leakage.
It is hard to tell if the amazon or similar cheap 9 V blocks are OK. Some are and some are poor. Expensive alkaline cells can also leak. If leakage is the main issue, long life Lithium primary cells may be an option too. I would have a slight problem with buying 12 batteries, as this would last for quite a long time and the shelf life is limited.