The problem is if the mint has high fee on inbound hops, or outbound on the next hop
Discussion
Yeah I don’t know how to check that if the mint is private other than testing a payment.
Probably they have a private channel with nostr:npub1jluy3twvf338v6zlujzzdhjkzjy8ezj34ksydr8vw8a6jwp89ygshpp2kq , or use it with blindedPath