First thing I can tell you is if the compressor is running, it's not your compressor or refrigerant charge - if the charge were low or the compressor had an issue, there's a shutoff device in the compressor to protect it or it sheared if it hard locked to keep the engine running - that's an intentional safety measure, as well as the shaft being breakable if it seizes. If someone mentions it, let them know you know they’re full of It.
I would check your defrost settings and see if there's an issue if you have not — a lot of Hondas use the same system for that as well. Put your old ECU in as well, especially if you have it and you have not properly programmed this one.
I would look at the A/C controls - pull the controls from the car with the fuse removed, and see if it still continues to start like that and the controls are NOT installed - if it doesn't continue, there's a short in the panel somewhere - if it does, that is fine but it may be worth trying a different one to be sure. The good thing is this is a Honda so finding it will be cheap and easy and there's really not a ton of risk that the part is NLA from the manufacturer. The Honda P/N is 79600-TM8-A53. The SM can be found here: https://lemon-manuals.la/Honda/2010/Insi...
We need codes and more testing to do more then give you this basic check list, with the old ECU installed. Other common things to check is the fuse box under the dash - replace it and see if it continues as it's a smart MICU box, even if it seems fine to rule it out. Also check for codes on the ambient temp sensors and data on ECT sensor 2 if you have the right scan tool, as well as the A/C request source data sources. Get it scanned with an all system scanner, or buy one that does bidirectional control with lifetime updates unless you think you may need ADAS calibration in the future.