Peter Reilley wrote...

Those cheap lasers are surprisingly good, between 1 to 2 mrad
divergence. This is the same for ordinary HeNe gas lasers.

That may be, but I have a few of these things and some of them diverge
noticeably more than others. Perhaps it's in the optics.

You cannot narrow the beam by passing it through a slit.

I disagree and offer the following for consideration:

http://www.paragoncode.com/temp/laser_beam.jpg
http://www.paragoncode.com/temp/laser_setup.jpg

I didn't place a ruler in the setup, but I believe the effect is clear.

A slit will cause the beam divergence to increase.

This is true, but like I said, within reason, the beam width itself can
nonetheless be decreased by a slit. To be fair, the definition of "beam
width" is debatable, but for practical purposes -- as in the OP's
application, it can be narrowed.

Jim