No, that’s not something Let’s Encrypt can do. They’re a public-trusted CA, and this request is to effectively allow you to issue malicious (even though your intent is not) certificates. That’s counter to the whole idea of Let’s Encrypt. I don’t expect that you’ll find any provider who would do this for free, or even for a low cost. The correct way to do this is to create your own CA which, you’re correct, would not be trusted by browsers by default, but assuming you’re MitMing machines that you actually have the authority to intercept traffic from, you would need to add your CA’s certificate to the trusted root store on those machines.
What you’re attempting to do is not a simple problem, and a much harder thing do do right. Given the level of experience your responses imply that you have with SSL/TLS in general, I would recommend in the strongest possible language that you hire an experienced security professional to help. In today’s world, you should assume that anything set up so insecurely will be broken.