I can see the company being paranoid, think about the Walmart crash a few years back where the driver was based at a terminal some 800 miles from his home and was allowed to commute in for the start of his work week without taking a 10 hour break before getting in the truck to drive. Driver was technically legal within the hours of service rules when he fell asleep and caused the fatal crash.
Given how LTL jobs work and the tight scheduling they may be holding drivers to living within a certain distance of the terminal as a way of shifting liability so they can say, hey Joe Driver told us he lived local, even had a local Ohio CDL, how were we to know he drove in from WV and was tired when he had the wreck?
Legally your CDL must be in the state you claim as home, so getting an Ohio CDL and keeping it when your legal permanent residence is in WV is a bad idea. Especially now that you know you can use your WV permit to train and test in Ohio (your company should be able to give a certificate to satisfy the BMV once you compete training and are ready for the skills exam), there really is no valid reason to have an Ohio CDL other than to satisfy some ridiculous company policy on state of residency to work at that specific barn.