How to connect unlocked iPhone 5s to Verizon
Unless you have a 6 or later, Verizon voice is CDMA-only for the 5s.

If your iPhone 5S is not CDMA capable, data might work on LTE portions of Verizon's network. It would still need to support all of Verizon's LTE bands (see iPhone LTE support). But, as mentioned above, voice would not work.  VoLTE is only supported on the 6 and later. 

Verizon has a database of iPhone IMEIs that are legitimately sold as Verizon "branded" or as world-unlocked phones, and that are CDMA capable. If they can't add it, there's a 99% chance it wouldn't work to your satisfaction.

You can test whether it would work at all by loading a Verizon SIM from a friend's iPhone 5S or later.  (A SIM from any non-Apple device won't generally work because there is a difference in back end programming, especially for voice + LTE.)

In the off chance that it does work on your friend's SIM, the trick is to get customer service or a store rep to load up a dummy IMEI, or the IMEI of a known compatible phone, just to activate the SIM. I've asked them to do this on occasion where I'm setting up an employee's account and don't have the phone present, but need it activated so it will work as soon as the employee reports to my office.

(For what it's worth, I find that AT&amp;T's lower frequencies in the 800-900MHz range tend to penetrate buildings better than Verizon or T-Mobile's 1900/2100MHz bands, at least in our neck of the woods in Utah. Your 5S isn't going to support the enhanced coverage offered by the 700MHz bands.)