Hi Nick.
Good work. The 402nd emblem is an early one from their days training on the P-47 in the USA (hence the 4 bladed prop); the 370th FG as a whole did their entire training on the Jug. Sufficed to say they got one hell of a surprise when they finally settled at Andover in the UK and found out they were getting P-38s!
In light of this conversion, the squadron emblem was redesigned to better reflect the squadrons association with the P-38.
I have only one image of the amended 402nd badge - it was, as were all the 370th FG insignia, unofficial (not approved by USAAF high command) - and comes from a photo of a hand-painted patch on a period jacket: