I will say, as someone who lives in a state with closed primaries it is very common for someone to register for the opposite party to vote for whoever you think will lose. This particular election cycle it would make sense that dude may not exactly be so much pro Biden, but anti-Trump, so he registered as a republican to vote against him in the primaries.
I am anti-Trump as they come, but it doesn't make sense that a republican would attempt to assassinate Trump. A lot of them may not like him, but they are a party of people that fall in line, not people that stand for what they believe is right. They are not selfless people that would give their life to protect the country from a facist. They are cowards that while they know it may not be right, at least he's on "their side". And let it play out.
You don't vote in your least favored party to vote for who you want to lose. You vote for who in that party you genuinely want to win - so even if your favored party loses, the winning candidate is at least not a piece of shit. It's an ad hoc way to do ranked choice.
This is why black voters in the south before the 60s frequently voted in the Democratic primaries despite the Republican party being less racist at the time; they could at least get an OK Democrat if they couldn't get a Republican.
Everyone should be voting in the primaries of their least favored party until ranked choice voting can realistically happen.
I mean a right-winger would certainly do the same to a progressive candidate to protect the country from healthcare.
But yeah a right winger targeting trump would need to be an anti-abortion or ultra-religious fanatic. Those are the things he's weak on in that world.
My money is on it being fame-seeking more than politically motivated, as it was with Reagan. That type of person is near impossible to anticipate so is more likely to slip in, I'd guess.