You don't need TLS for that, you could simply use an HMAC and a shared secret, assuming you're not worried about people with physical access (and ability to get the secret) being able to create updates. Of course if you've got multiple instances of the device (not some hobby thing where they are all owned by you), then the secret for each device should be different so someone can't buy the device, determine the secret and then push updates to other people's devices.
RSA means big integer which means unhappy performance on devices that often don't even have floating point in hardware. I think elliptic curve could be faster?
