Getting EVA foam to shine is a heck of a process. I have tried working with automotive paints in the past, and while those spray primers are appealingly easy to use, I have never been able to get a smooth, glossy finish when using them. Part of the problem is that you need a ton of layers of the primer in order to create a smooth surface, but there's also the matter of most automotive primers not being shiny enough.
Instead, I would recommend using a PVA-based primer. Some people use white Elmer's glue (the stuff for wood, not the washable stuff for little kids), but it isn't as flexible as some options. If the helmet isn't squishable, due to internal structure or something like that, then that isn't a concern, but one of the main draws of EVA foam is that it does flex. So, if you need it to flex, that might not be a great option.1
I prefer to use Flexbond, which is a primer designed for theatrical backdrops. It can't be sprayed, but it is largely self-leveling, especially if watered down the appropriate amount. 2 I've found that Flexbond works really well for giving a high-gloss shine.
If the high-gloss spray paint works over the new primer, great! If not, I recommend finding a good glossy acrylic. (There's a line of paints from Plaid specifically aimed at cosplayers that are gorgeous, albeit expensive, and they're very prone to sticking to each other, but we'll solve that in a second.) You'll also want to add shading along the contours of the piece-- highlights at the edges and shadows in the depths. This step is huge, and goes really far towards tricking the eye into believing that an item is shiny and metallic.
Finally, get a good high-gloss top coat. This is the only part where I prefer something sprayable over something brushable (but that's just because my airbrush is currently out of commission, so I can't use it for the acrylics). This serves two functions. The first is to protect your piece from dings and scratches, and also from sticking to other painted pieces. 3 The second is that, well, it adds a bit of extra shine!
1 Spray paints also don't flex, and you'll probably wind up with cracking over time if the helmet does get bent or squished in any way. Acrylic paints are much less likely to crack like that, as long as they're decent quality.
2 I like to make mine the thickness of those probiotic drinkable yogurt shots-- just a little bit runnier than it is out of the bottle. Undiluted, you can get clumpy drips along the contours of the piece.
3 Seriously, this is not a corner to cut. I have made this mistake so many times, and wound up in tears because my paint job wound up ruined.
I also want to offer a disclaimer that I'm not sure that any of these techniques would hold up as paint jobs for actual LARP combat, but given that the helmet is identified as being a "prop", and given the techniques and materials that have already been tried, I suspect that the goal isn't for this helmet to be taking blows.