First, it is definitely true that a flat, un-matured beer will seem light in the mouth feel in general... so that in itself is not a major indication. Even though a carbonated beer is technically less dense the CO2 will by it's nature will increase the mouthfeel.
SG 1.037 / 9.25Plato is a relatively low gravity beer and won't be very dextrinous no matter what you do.
What was your grain bill? I assume that, being in the UK, you probably used a British malt for your base. If you haven't already been using it, you could try adding some cara-pils to your grist.
Shortening your mash will help. If using a good, fully modified malt, 30 minutes is plenty. I do 30 + maybe a 5-10 minute vorlauf to clear my wort, then run off to the kettle. Whether in a commercial brewery or at home I never go beyond 60 minutes.
Not sure which yeast you are using ( nor did I take time to figure out your projected degree of apparent attenuation, I'm lazy! ) but if you use a yeast that is lower attenuating you will obviously have more mouthfeel in the end.
As far as doing anything to THIS batch now that it's in the primary, I wouldn't... let it be. The beer will be good, you will enjoy it and even if it isn't 100% exactly as you wanted it you will be able to use it as a jumping off point for your next one.
"Why did you.... what was the point of... how drunk were you when you decided this was a good idea?" - DMartin