10 minute window of confirmation for a bitcoin transaction is just dumb and it has nothing to do with fees, backlog or anything else. 10 minute is the average time when a block can be found. sometimes it is 1 second and sometimes it can go as high as 1 hour.

what websites that accept payment like this do is that they have a X minute payment window not "confirmation" which means you have to make the payment in that time frame or it will be invalid. and if you do then they will wait for it to confirm which can be longer than 10 min.

ps. have you contacted the website with suggestion to implement LN?