I've thought about fitting a pressure-equalising valve
(one for the whole bathroom, rather than one per shower),
such as https://www.plumbworld.co.uk/344-0000.

Don't even think about it. Pressure equalising valves are great, but their
purpose is to put both supplies on the lowest common denominator. If one
supply is poor, it makes the other equally so. You first need to get two
good supplies.

The correct solution is to put a single impellor pump on the hot side, to
boost the hot water system to mains like pressure and flow. Then you can
worry about pressure equalising, although this will only really be
advantageous on a shower mixer, which a good model will have built in (along
with thermostatic operation).

An alternative to the pump is to fit a "venturi" type shower, which uses
energy from the cold mains to boost the hot water through the mixer. This
won't help other outlets, like the bath taps, that might also have poor
flow.

- I'm planning to put a double-check valve on the CW input
to the valve as well - is this necessary, or could it cause
problems?

It is required for hosed shower mixers unless you put one of those intensely
annoying clips on the shower hose that prevents you using the shower to
clean the bath/cubicle with.

Christian.