Some more thoughts. A 3 bar pump will supply about 50l/m
with a head of 10M ie close to 1bar, that's more than enough to drive a
shower head and a few body jets IME. However, the flow rate at
1bar through a 1.5mm jet is about 1.5l/m so if the body jets have 40
holes, you'll need an additional 60l/m to feed them!! In all
probability, the body jets have about 0.8mm holes, which would reduce
the flow rate to say 17l/m. So it looks as though the problem is to put
40 to 50l/m into the pump. At 25l/m flow, (considering one flow
only--hot) you will lose at least 1.5m of head(head in this case is the
height of the water column feeding into the pump) if only 22mm pipe is
used, if you reduce this to 18mm, you can't IM estimation achieve this
flow rate. If you go to 28mm, the available head at this flow rate
should improve by about 1.2m, giving you a real chance. Just another
thought, the pump is mounted as close to the floor as possible isn't
it?, because this will increase the head of water available and improve
the flow rate into the pump.

I have just measured 45l/m coming out of a not very free flow
bath mixer tap with a pump on and a 22mm check valve in series(o/p side
of the pump), so your shower should be perfectly feasible.
You might have a Google at "watermill showers" and look at
their installation instructions to see if it helps.


Regards
Capitol