I'll point out a few other things:
Per FCC rules, the licensee is the only one who can add radios to their system.
§90.427 Precautions against unauthorized operation.
(a) Each transmitter shall be so installed and protected that it is not accessible to or capable of operation by persons other than those duly authorized by and under the control of the licensee. Provisions of this part authorizing certain unlicensed persons to operate stations, or authorizing unattended operation of stations in certain circumstances, shall not be construed to change or diminish in any respect the responsibility of station licensees to maintain control over the stations licensed to them (including all transmitter units thereof), or for the proper functioning and operation of those stations and transmitter units in accordance with the terms of the licenses of those stations
Translation = The person/company whose name is on the FCC license is responsible for all radios used under that license. That includes your radio if you are using it as part of your job. You, as an individual, do not have an FCC license for those frequencies.
There's nothing wrong with what you want to do, just make sure you get something in writing from your employer that says you are permitted to have your own radio operating on their radio system and under their license.
That also means that your radio needs to meet all the requirements that are set under their license. That means meeting emission requirements, power levels, usage locations, etc.
Not a big deal, lots of people do it. Just make sure you have the paperwork in hand, and that your radio is fully up to snuff.