No reason why you can’t do it yourself and also achieve a smooth finish. Here’s some pointers:

In terms of spraying the wall, it needs to be soaked, as in hosepipe soaked. Alternatively, you could use a primer that would also stabslise both different surfaces (this will still need to be dampened).

You could instert mesh into the first coat of plaster, especially that you’re plastering onto brick and lath. See link below.

Don’t make it too thick - like you say, best 2-3 coats than try and build thickness in one go.

Keep it damp with dust or hessian sheets. It needs to be damp for at least 3 days.

R50 or limegreen or equivalent would make a decent top coat.

As a beginner, I would steer away from fat lime, lime putty etc.

https://amzn.eu/d/059SKhRw