You can, but cutting leads flush after soldering stresses the joint significantly. Cutting them prior to soldering means you may not be able to get a good solder joint, requiring either re-soldering or 10x visual inspection.
Since the battery and holder have significant mass, the strength of the joints is important so you might want to do some testing (such as drop testing).
This is not standard handling, so it will be up to the PCBA house whether they want to do it and how much they want to charge.
Read More: Through Hole PCB Assembly