There used to be a friendly hostel downtown called "The Bent Prop", but it seems to have gone out of business during the pandemic.
My guess is that you will not have any problem finding bike-friendly accommodation in Anchorage. Alaska is full of pretty adventurous and atypical people, it's pretty unlikely that anyone would object to you spending a few days getting your bikes organized for your trip.
Be aware, though, that Anchorage can be a bit of a rough town. It's a problem that is common in all remote areas - the biggest town in a remote area tends to be where people who are down on their luck eventually wind up. So pay attention to the security of both yourself and your belongings.
Michael
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