Anyway to bypass the M3 - M4 bottle neck ?
Yes if you want to go via Birmingham... A281 to Guildford, then A31, A331, M3, A34, M40...
I did actually think of that way
I was down in Abingdon a few weeks ago and that the way I went.
I don't mind a few more mile, its the sat still that does my tits in 
At the end of the day, the company pays for the fuel 
You could do Birmingham, M40, then M42 or M69 but its a bit of a fuff around, I used to do London, Hull and back on a regular basis a few years back tried all the routes many a times, in the end even though Im only 5 mins to the M11, I would always go M25. M1, M18, then north side of the Humber into Hull, always the quickest way. Theres just too many Roundabouts and HGV's on the non M parts of the A1.
Problem your going to have TG is leaving Newhaven at 7am, your going to hit the M25 peak rush hour(s), and I wouldn't go anywhere near Heathrow territory between 7 and 9.30 am.
Think if was me, I'd be having a lazy Breakfast, then aim to hit the M25 at around 9:30, M25 Dartford to M1 etc,
No easy way really, Google Maps with Traffic Update on will be your friend and fingers crossed..