Hi - just my ´two-pennies´ ...
Quick calculation: 380 mph indicated (x 1.609), is 611 km/h but at 11.000 feet translates to true airspeed (TAS) of 745 kmh but the GCI mesurement of 458 mph, groundspeed comes out as 736 km/h (he then had an slight headwind or crosswind component of about 10 km/h ~ 5.3 knots).
Possible piston aircraft can be Ju 88 T-3 with GM-1 or Ju 388 L-0 / Nacht - sufficent quantities of both types had been produced at that time - and delivered to the Luftwaffe - but who knows ?
Likely not an Ju 88 S as that was the Bomber version, but I remember reading on Ju 388 performance ´The maximum speed is 383 mph at 40,300 ft´ in the Vernaleken / Handig ´Ju 388 book´ or
ttp://www.ju388.de/
That only makes (616 km/h) but finding exact performance at 11,000 feet (3.300 m) was harder. Same book P.181 has performance chart of Ju 188 J-3 with various engines but none reaches that at 3.300 m in level flight - just possible in a dive (and 770 km/h were recorded on Ju 388´s).
Is it possible this plane DIVED below the overcast (as stated abowe) - to take photos - reaching that speed whilst in the dive - levelling off - then the Mossie was able to chase and close in - but then outclimbing the Mossie in time - as Ju 388 L climbing ratio was indeed excellent! But then likely an T-3 if it can match that speed.
cheers
ed