It has to be sorted out for each state. Some proportion on where you reside, some on where income is earned. A domicile state might want taxes even if you do not reside, work, or get paid there. I live in a state where domicile = residence for tax purposes, and have previously lived in a state where what mattered was where the income was earned.
How much time you can be in a place without being a resident varies quite a bit also, and it is different whether or not you are working in that state, or living there and working in another state. I suspect your employer may find this solution even more trouble than having you live in Idaho; businesses tend to locate in Florida to avoid the cost of dealing with state income tax.
I know, have worked with, a number of people who work as consultants, some internationally, some just around the U.S. They have various arrangements to deal with income tax issues, some based on "you are paying my firm for services, where it is headquartered" but a lot of what makes it work is that the job is only a few days to something short of two weeks. Even then, the task often gets booked as a "business meeting" local rather than a service. I know some who maintain multiple part time residences in states with particular tax laws and have another in an offshore tax haven, do most of their work in places they don't reside, and I don't have a clue about their domicile, but do know they have good professional tax help for guidance.
If you really want to live most of the time in Idaho, and your employer doesn't want to deal with the tax problem, I suspect your employer will not want to deal with a messier problem of moving from state to state at intervals. A really good tax attorney could probably work out a plan, but it likely would not involve living any significant length of time in a state that proportions income tax based on residence.
