Looks good, just some nits around testing.

I think it would be good to have some autogenerated unit tests as well so we can keep generated code coverage at 100%.
Maybe something that injects a dummy transport that can tell us that it was told to enter, something like (strawman expressions in the template, use the correct ones :P )

class DummyTransport:
    def __init__(self):
        self.trigger_state = 0
        
    def __enter__(self):
        self.trigger_state = 1
        
    def __exit__(self, *args, **kwargs):
        self.trigger_state = 2
        
client = {{ api.module_name }}.{{ service.client_name }}(transport=DummyTransport())
assert c.transport.trigger_state == 0
with client:
    assert c.transport.trigger_state == 1
    
assert c.transport.trigger_state == 2

And then associated mox tests for the grpc and rest transport enter/exit methods (I always have to steal mox setup from existing code).

Looks good, just a few more obnoxious nitpicking comments :)

Thank you for implementing this!

CC @tswast @nicain I think both of you were interested in this feature, PTAL if you have a moment.

The added context manager method looks good, just one minor comment.