This is a Frame Challenge
How do you defeat a godlike character? You can't. You must introduce a weakness, an Achilles heel.
You've created a "godlike character" by developing rules of time travel that forbid proof of action. There's some problems with it...
Apparently the suit is destroyed only after the round trip, which doesn't make sense. If it can be designed to survive traveling through time twice, it can be designed to travel through time thrice, thereby eliminating the "you're nekid!" restriction.
Ignoring Arnold in The Terminator, the idea of everything "not you" not surviving the trip is ridiculous. Does hair survive? If so, then will anything made of Keratin survive? How does time travel know when something is "part of you" and not "part of you?" In other words, since you're inventing a non-science-based rule, who's to say they can't slip a piece of paper inside the suit and travel back with it, the suit disappearing and the paper, now unrecognized by the traveler, falling gently to the floor? No suit is actually "skin tight..." so how would your system even know if something not-the-traveler is inside the suit?
Are you saying the amnesia only happens on the return trip? So the traveler is fully conscious of who they are and what they're doing in the past? What is it about the return trip that does this but doesn't do that after the first leg?
In short, you've created a godlike problem and it can only be solved in the usual "how do I defeat my godlike character?" way... you need to introduce a weakness.
So, in regard to my original answer, below, while it might be problematic to slip the Colossus of Rhodes inside your suit, I suggest that as a necessary component to solving your problem, travelers can slip things inside the suit... like a small Roman Fresco.
Air in their lungs
But, if that's not your cup of tea, the problem is automatically solved by analyzing the air inside the traveler's lungs. Unless you further perfect your godlike machine problem by stating the suits are hermetically sealed and the travelers can't leave their suits at any time.
This was my original answer before thinking about the "can't carry anything, can't remember anything" condition. It's here both for the Frame Challenge, above, and for entertainment purposes.
Take a piece of artwork believed to be missing
There are a fair number of pieces of art that have become lost to history.
Turns out at least one of them was lost to history because your time travellers boosted it.
The art can be proven to be from the past because it's missing all the chemical hallmarks of existing in the atmosphere during all those years. In fact, the further back you can go to (*ahem...*) borrow a painting the better due to the decreasing number of pollutants as you travel further into the past.
Who would have thought that polluting our atmosphere would prove useful?
While Roman frescoes would likely be the most practical choice, since you can travel a long, honking way into the past... long before modern hydrocarbon pollution came into play, and therefore prove you didn't just skip over to Italy and boost a piece while no one was looking... I should think you could get extra points for showing up with the Colossus of Rhodes.
Timmy... how'd you get that through the portal?