FWIW, Budget ANC is indeed a franchise and is owned by the local
Kendall car dealership. I've rented from them several times, but I also knew several of the employees/managers there (no more, though--I think they're all gone), so that was probably my free pass to avoid problems.
State law does not
require you to purchase
SLI (best to use the term Budget uses for it; LIS is Hertz's term for the same coverage) to rent a car, but Alaska is relatively libertarian and doesn't have any consumer protection laws on the books (AFAIK) that explicitly restrict the rental companies from requiring the purchase of coverage (several other states do). That said,
a complaint to the state Attorney General may be fruitful, and I would echo the idea to file a complaint with Budget corporate as well. Be concise in your complaints, though: all the fluff about bouncing between counters and all that is immaterial; a simple two-sentence statement such as "After being forced to wait one hour to pick up my car, the agent refused to rent me the car unless I purchased Supplemental Liability Insurance, and a manager also refused. I was ultimately able to prove and convince the staff that I was not legally required to purchase it, but this significant hassle and coercive attitude reflects extremely poorly on the Budget brand" is all you need.