Yes, this.If your ground units can hear the repeater at all times, increasing repeater power won't help. It appears your problem is the repeater hearing your ground unit handhelds. The fix for that is get the equipment a tune up, make sure the antenna, feedline, duplexer and receiver are working top notch (why leave performance on the table), then if you are replacing you need to get the antenna higher or add additional receive sites.
You're experiencing a 'talk-in' problem, not a 'talk-out' problem. And high site noise or site noise generated by the transmitter energy impinging on nearby rust junctions, etc (and this is just 1 of the problems that can crop up), and many other factors can kill your talk-in sensitivity.
"Repeater Site Sensitivity Test" is the phrase your looking for.
Yes, some basic tests of repeater RX Sensitivity, TX power & modulation, VSWR, site sensitivity, antenna & feedline sweep would give you an understanding of your current baseline performance. That will give you a chance to know what kind of improvement you really get with the new equipment.
Others have said it right, if someone in your org tells you something can't be afforded, you need that answer in writing. Cover year... head.
(You're in for a slog, friend, and if it doesn't work out, you might have your resume updated and circulating for a quick exit. )