they have a 10 minute timeframe for the transaction to confirm.
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?
Yes, I did contact them about implementing the Lightning Network, but they have not seen anyone asking for this, so they reckon if there are not a lot of people asking for it, then it is not worth the time and effort to implement it.
I did ask them to increase the 10 minute confirmation window, but they say the volatility will eat into their profits, if the price goes down, like it is doing at the current moment.