Although the Brown Pneumatic pistol in your picture is not in the best of shape, it is indeed the Deluxe version of an extremely scarce American air pistol from around 1939 (you might google "brown pneumatic pistol" for more info). A somewhat cleaner example of the standard version, also with its box, just sold at Holt's auction house for £3,000 before fees. If you were to sell it to a collector directly you will get far more than the dealer's £300; I'd expect a price of £1200 would bring a quick sale.
Please note that my frame of reference is here in the USA where there's not as much airgun collecting interest as in the UK, so my estimate may be LOW).
Don R.