Can bike = permit be used to transport information to outdoor app users

The whole reason permit was invented is because of US national parks: you don’t have a legal right to access them, so yes would be inappropriate, but there’s a permit system and a permit is often quite easy to get, so mappers were looking for a less restrictive alternative to private.

  1. Is it correct to use bicycle=permit in your case?

There are different points of views on this. Some mappers use =permit for every path that requires a permit. If you take that view, then it is correct. Other mappers reserve =permit for paths where a permit is relatively easy to get, like in the case of the US national parks that the tag was invented for. They insist that if a permit is hard to get or limited to specific groups of users (e.g. residents only) then =private should be used.

  1. Will it result in a warning message?

Unlikely. A lot of routers treat permit like yes. Last time I checked, only GraphHopper’s default profile treated it differently, but it refused to route along =permit entirely.

If a permit is hard to get then you might want to use “private” instead. Many routers, including Komoot, will not route along a private path at all. OSMAnd will show a warning if the destination is private.