This is very complicated I think, especially it's an online business since Bitcoin transaction takes time before you receive it. But if you really want to apply it, I think using cold wallet is better since you're going to receive from different addresses. It's indeed profitable in the long run but I think customers will lose a lot since Bitcoin is very volatile and spending for example an $8 product plus transaction fee right now would be more like $16 when Bitcoin's price pumps again.
Anyways, this is just my own opinion.
I would also think the same, because if the site is hacked, it is most likely that the Hacker will look for all the currency that has to do with Bitcoin, and that is what should be protected the most, for me these things have a lot to be careful about, It would be great if the entire platform could be managed under LN, it's the best, and it could also give that the fees are cheaper and everything, obviously this is a suggestion, but I like the Cold Wallet as you say, security above all , because if they are capable of exploiting sites like casinos, exploiting them all with a simple vulnerability, now an online site is easier for a hacker.