We put our 2008 lift (a 118 inspiration) on a kitbot and attempted to pick up tubes. Our claw is a bit rough but definitely helped us decide on the roller claw vs. regular claw debate.
Using omni wheels for driving is likely to severely limit your traction/pushing power, but if you think the extra maneuvering is worth it, that chassis is an option. However, I think mecanum wheels will get you that kind of translation with much better traction.
We got lots of our manipulator ideas by looking at the successful robots in Rack’n Roll, because this year’s game pieces are very similar. The roller claw CAD design you linked us to looks very similar to the one that team 257 (I think) found great success with.
I’m interested to see which design you finally choose on. Good luck!