There are two issues here. One is what will enhance your future chances to get in to a program, I assume in the US. The other is burnout. I'll address them separately.
Burnout is a serious issue and you should take some effort to avoid it. But working in a research project isn't, I fear, different enough from what you have been doing lately to let your mind settle down and relax. I would suggest that a few months of travel (traditionally the South of France or Spain) would be a better solution. Get away from things. But 18 months is probably longer than required for this. Walk the Appalachian trail. Ride a bike from NY to FLA.
The other issue is about enhancing your application. In Europe, some research experience would be helpful because the European baccalaureate is much more specialized than one in the US, so more is expected of entering students. But in the US that isn't normally necessary. The doctoral program in the US begins with mostly coursework (or what I characterize as "warmup" research projects) and not the serious dissertation research seen elsewhere. Not every entering student will have any real research experience whatever, and little in the beginning curriculum requires it. There is a lot of coursework to enable you to pass comprehensive exams in the field. This is the "breadth" element of graduate study, followed, after a few years, by the "depth" element of dissertation research.
So, while a serious stint of research might offer a small benefit in admissions (in the US), I doubt that it is necessary.
But your topline question is whether it is possible to do this. I suppose it is, but it takes time and work to find a situation. A lot of places doing serious research will think you are unskilled, not yet having an advanced degree. You will be a "trainee" for a while. It probably won't be very satisfying.
Some people, however, find a way to stay on at their undergraduate institution and do some research with a professor or two, but otherwise only carrying a light load. This can be the best situation if you can arrange it. Another possibility is to have a current professor connect you with a colleague at another local institution to do something interesting. This might be a sufficient change of pace to avoid burnout and also keep you connected enough to academia to make for a strong application. Financing this might be a problem, unless someone has grant money.