Am I searching for a hens tooth?

[Geezer]

Aeron-Brno B96 and B98 are another, though scarce, option.

[Modski66]

Bit of a curve ball, but how about the Umarex S&W 586? Apart from it not being PCP, it seems to tick all your other boxes - self indexing, adjustable sights and a reasonable trigger.
I am not a great fan about the looks, but I have great fun shooting with mine - especially with the 0.22 9.5 inch barrel fitted.

[LewisYorks]

I don't know of anything you could get for 200-300 quid which ticks all your boxes.

I have a HW44 which is (I think) the ultimate plinker but it's not semi auto and you said it's too pricey.

I also have a Steyr EVO10 which is great again but again not multi shot and out of budget.

I would just stick with the usual CO2 plinkers or raise the budget and go for the Steyr you mentioned.

Lewis