"Our custodian would freak if he had to continually clean up dropped fruit"

Tell the custodian to read Matthew 21:19 and Luke 13:6-9.

Then plant something that does bear fruit ;-)

What's best for you will depend on the location, which you've not given.

Resin

Sounds like a flowering crabapple may work well. These tend to remain on the small side as trees go, flower profusely in spring and often provide attractive fall color. And while they do produce fruit, it tends to be persistent or remain attached to the tree until eaten by wildlife or falling off in spring. The fruit is considered a very ornamental winter feature for most cultivars. But they are far from messy.

I'd recommend Flowering Dogwood, ... as long as your area is not having issues with Dogwood Anthracnose.

Dogwood is also a spectacular Spring bloomer, with a longer blooming period than Crabapple. Dogwood has red foliage color in the Fall, which also lasts for, at least, a month.

Crabapple is a good choice as well, but there are some cautions ...

1.) Make sure that the cultivar that you plant does not drop its fruit. Many of the older cultivars dropped acorn-sized fruit (exactly what you said you don't want).

Many of the newer cultivars, however, hold on to their (smaller sized) fruit through the winter - until the birds eat them.

2.) You might want to reconsider Crabapple if you have any Juniper plantings.

Crabapple planted in the vicinity of some Juniper/Cedar can develop a fungal disease which will affect its foliage in the late summer.

3.) Crabapples are subject to any number of plant disease which can dimish their attractiveness. Be sure to plant a disease resistant cultivar.