Rspected, I noticed you also copy & pasted the identical I3-swap opening message onto another popular Emira forum, and also as your first post there.
Whether or not you wanna admit it, you seemed to be looking for a bit of drama to kick things off

. If you have the car experience that you say and this is not your first rodeo when it comes to car forums... Then you know cross-posting a message as your first post on multiple forums where you suggest throwing out the existing powerplants for the Forum's car and replacing it with something from a different manufacturer (which is significantly underpowered by comparison) isn't likely to gain you a positive response. But as you know, it WILL generate a response. Simple introductions might work better and certainly most in your position would have chosen to lead with those pictures of events you've attended and stating their interest in the car.
As mentioned by others, basic etiquette on any forum, sports car or otherwise, would also be to take some time to observe the forum and get to know the subject matter a little first. Even if you still felt that somehow you wanted to discuss an I3 Toyota motor on a Lotus Emira forum (actually, on 2 different Lotus forums), you could still approach it by identifying why it would make sense over the obvious comparison of what already comes on the car. The 2.0 turbo I4 engine in standard Mercedes production form already makes 416hp and 369lb-ft of torque (Merc A45S) and the aftermarket has tweaked them to 500hp. Doesn't take more than a few mins of searching to find that.
Now, that aside... Welcome to the forum
As for the question of aftermarket impact on cooling.. A lot of aftermarket parts co's should have successes with this on the I4. I'm not sure there's an answer yet (?) on how Lotus will treat the gas particulate filter (GPF) for the US market, it's a giant heat radiator taking up a lot of space and sitting right by the engine near the top of the engine bay. That's in addition to whatever cat converters are also in the engine bay area. Removal of some or all of those items and wrapping/heatshielding the exhaust piping after turbo exhaust manifold exit will have a significant impact on dropping underhood temperatures. The reduced backpressure will help too, and the turbo response should notably improve.
That said I don't know how many total cat converters will be on the I4, and which will need to stay to prevent a check-engine light, and/or if a higher-flow aftermarket replacement might give performance improvements without generating a light. Or if a simulator piece or spacer piece can be used to make the ECU think the stock cat is still there.
Many of us (myself included) still have to wrap our heads around a car where the ECU will be locked up tight and the manufacturer is set up very hostile to any tweaks to it.