I agree with Jim, the LBS answer seems calculated to make you just buy something. Thing is, are they skilled in vintage components like cup/ball BBs specifically for Campagnolo? What kind of Record is it? C-Record has some unique features making it different from Nuovo and Super Record.

And while 115.5 may be a decent approximaiton of the necessary spindle length, it is not a symmetrical BB, and it's usually for triples. From my refurbishment of my 1980 Masi with Nuovo Record, I know that when it's assembled correctly, everything fits the frame and the BB like a glove with the correct spindle, cups, BB shell, and crankset, and the drive and non-drive side location of the crank arms are symmetrical with respect to the chain stays. I would want to spend a lot of time in trial assembly to verify whether the Veloce has the same dimensions as the appropriate cup/spindle combination.

What I would do practically with your bicycle is first try to adjust the existing bottom bracket for the correct play (zero play but zero extra friction after tightening the lock nut), and then do a few days of riding to see if it works ok. Then measure the chainlines, Q 's (both left and right), crankarm clearances, and other possible measurements to see if it's right, then doverhaul the bearings to check that the bearing is aligned ok internally, then with all that data assess what it really needs if anything. If the spindle is really ok with the Campagnolo crank you are just lucky IMHO, but it might be ok.