A cross-party delegation has no right to speak to the EU, even supposing that OG is wrong and that such a delegation would work (although why he's any more sceptical about that than about May et al's ability is anybody's guess).
Only Government -- or, really, The Crown -- has the authority to conduct foreign affairs and treaty negotiations. Parliament's role is to approve and to ratify. The best a cross-party delegation can do is to deliver to the Government (ie, to Theresa May) a set of proposals that she then negotiates.
Constitutional law is messy, but a lot of this is indirectly set out in the Supreme Court ruling on whether or not Government had the authority to unilaterally withdraw from the EU (ie, to unilaterally give Article 50 notification) -- the answer was no, because in that case doing so would also have had the effect of repealing domestic laws, which the Crown (Government) can't do on its own whim.
Does that clear things up? Probably not. But basically foreign affairs are for Government and domestic affairs are for Parliament.