The easiest way, although not the least expensive, would be to add a Naran Push Microbot. This is essentially a tiny robot finger and all it does is push a button on request. So it’s a very simple way to retrofit an existing button remote. It works quite well, and I use these in my own home.
The main problem is the cost. Each Micro about cost about $49, but then you also need the $89 Prota bridge in order to be able to use the IFTTT channel. Each bridge can support about three dozen microbots, but it’s still an expensive solution. So good for some situations, but I’m not sure it fits your use case because of the cost.
If you happen to have a Harmony hub, it’s possible that your LED strip is compatible with that, and if it is, it’s easy to add an activity that can control the lights. I use this with some flameless candles, for example. Again, that’s an easy, no wiring solution, but it will only work if the remote is either an IR remote or a Bluetooth remote.
If it is an IR remote and you don’t already have a Logitech Harmony have, there are some other less expensive IR remotes where you can get compatibility with SmartThings, in particular the Zmote.
As far as solutions that will require taking the remote apart and rewiring it, I will leave that to others to discuss. 