By law, it seems I can't get residency until 90 days. What can I do? I can't get a plate until I'm a CO resident.
According to the state,
If you own or operate a business in Colorado or, are gainfully employed in Colorado, or reside in Colorado for 90 consecutive days you are a resident of Colorado. After becoming a resident of Colorado you must register your vehicle within 90 days or you will be imposed vehicle registration late fees.
"Resident" is defined in C.R.S 42-1-102(81). So one solution would be to get a job within the state.
You have to have your vehicle registered somewhere to legally operate it on public roads. If Colorado won't allow you to register it, then you should renew your registration with whatever state had previously registered it.
You could also consult your local state representative's office: They often help resolve these sorts of conundrums created by state laws.
As a practical matter, since you probably also need to have insurance to operate it on public roads, and since auto insurers are experts in the tricky area of interstate vehicle regulation, you could consult your insurer for guidance on getting your vehicle legally back on the road. (E.g., just say, "I want to cancel my policy unless you can figure out a way for me to get my car registration renewed." Insurers don't make money if you aren't paying for coverage.)