Iam installing BR 111 3/8x3" engineered, BR111 calls for 3/8x 1 1/4" staples.
Local rental shops have cleat or 1'x3/16 staplers only.Subfloor is 3/4 T&G plwood.
Any recommendations/ suggestions. Thanks in advance.

A 3/8 to 7/16 thick floor is commonly stapled with 1" staples. Once you jump to 1/2 to 5/8 thich, you will need the 1¼" staples of the Spotnailer brand floor runner.

As a test doing 1/2 engineered, I used my Senco SLS-20 to see how well it held with the 1" staples. Well, it held really well, and you had to destroy the tongue to get it up, prying with my hammer claws. The staples stayed in the subfloor, with splinters of wood attached.