there is no 6-speed compatible with the Omega 54-degre V6 without making stuff to suit. forget adaptor plates, etc. they're a real bodge.
but when you go to LS-power, the easiest is to use the T56 that'll come with the engine.
actually, there might be a way, but it'll be really expensive and not strong enough for a car of the Omega's weight - you'd use the Caterham bellhousing for adapting a 2.0 XE engine to Caterham's own 6-speed box. BUT the box is a modified Sierra type-9 box and probably not up to the job. expect bellhoisung + box to be more than the cost of a LS1 + T56
ISTR the Caterham 6 speed has daft ratios, designed to keep a 1.4 K series on the boil, as well.
6th gear is 1:1 and 1st is quite high, certainly <3:1. So you'd need a diff from a diesel to get decent RPM at cruise and a bootload of spare clutches for pulling away. In-between those 2 extremes your left hand will be a blur.

Which brings me to the point I was going to make:
Why do you (think you) need a 6 speed gearbox?
The GM V6 in standard form has a wide power band. There's good torque from 2K RPM. It's certainly pulling well by 2.5, all the way up to 6K+.
With a 5 speed box after changing up from each gear you will be landing back at an RPM where the engine is well and truly "on the boil" so why do you need another gearchange? It will probably just slow you down.
If it were a highly tuned engine or a diesel it might be a different story...
Kevin