Best method of bouncing laser off series of mirrors

What you ask is accomplished on a regular basis in a different plane for laser cutters, specifically CO2 in this example. It can certainly be applied to an LED laser and is often done so in the cutters with a beam combiner, used to assist alignment.

The above image is from an Aliexpress product post, but is consistent for many laser cutters. Variations of this design also exist but have a similar foundation of construction.

The base provides for gross adjustment, an eyeball alignment of sorts. The mirror fits into the circular cut-out, barely visible in the image. Missing in this image is a threaded ring holding the mirror disk in the fitting.

Once the gross alignment is accomplished, the knurled screws (and knurled lock rings) and adjusted for fine tuning. The spring-loaded bolts keep the mirror plate against the adjustment screws and can be adjusted for tension.

In the world of reflection, the angle of incidence is equal to the angle of reflection. Incoming laser beam at 45° to normal will strike the mirror and exit at 45° to normal. A very small adjustment at the mirror will result in a large change in position over a distance as great as described.

For your application and ease of adjustment, you could use a bright flashlight in the laser position for the gross positioning in a darkened room, then replace it with the laser for the fine adjustment.

If determining the laser position is difficult, holding a sheet of paper or other obstruction in the path is also helpful.