Your LBS is technically correct in that a 68mm BB can be used in a 73mm shell (given the threading of cups and the shell match, ie: BSA threaded BB in a BSA threaded shell). There are many reasons to NOT use a 68mm BB in a 73mm shell. In many cartridge bearing BB's, the right (driveside) cup is fixed to the cartridge (bearing assembly) and is also flanged. That acts a stop. When a designed 68mm BB is placed in the 73mm shell, it will be shifted, relative to the frame centerline (which is a point from which the chainline is determined) 2.5mm to the right, driveside. This will move your front crank's chainline 2.5mm outbound. The chainline increases by 2.5mm. Front derailleurs work best with a chainline between 47.5-50mm. Question becomes can the derailleur move out enough to get the chain on to the outer, large chainring with the wider chainline? Smoothly and consistently no less.
While that is being considered, at the non-drive side, the left crankarm is gonna be 2.5mm closer to the frame since all things to do with the BB have shifted 2.5mm toward the drive side using the shorter, 68mm BB in a 73mm shell. Question that will need clarification before riding is, does the left crankarm have adequate clearance of the frame now (specifically the non-drive chainstay)?
Also at the left in this situation, is checking to make sure the left cup of the 68mm BB can thread far enough into the 73mm shell to adequately support the cartridge bearing--spindle assembly AND adequately preload the bearings. While many left/NDS cups come flangeless and can be threaded past the outside edge of the shell, the splined tool fitted into the splined outer aspect of the left cup needs adequate engagement to reach proper torque. In addition the cup may reach the end of the shell threads before the system has adequate preload.
In short, there's so many gotchas, that it becomes not worth it. Regarding the discrepancy of the spindle length of the proper, 73mm BB, it's best to go a little longer. Again, you may change the chainline with this approach so you don't want to go a lot over. Shimano does make a 73/113 BB-UN300.
One hack you can do (and this is not the best thing) is using the cups from your current 73mm BB. You'll have to remove the non drive side cup and the old spindle and bearings. Then, take your new 68/113 BB and remove its right cup off the spindle/cartridge bearing assembly which you can then insert into your shell (mindful of the correct orientation of it). You may have to tap it into the right cup of the old BB which is still installed in the shell. Secure with the old left cup. This hack has its pitfalls as well, including possible damage to the new cartridge. Also if the new BB's right cup is fixed to the cartridge with more than a press fit, you likely won't be able to remove it without damaging the cartridge. So....