I have a case in which everyone agrees a change of venue is warranted. If the judge transfers venue (most likely outside the judicial district), can the defense preclude the judge from then presiding over the case simply by refusing to agree to him going with it under 31.09? Does it make a difference if he transfers it on his own motion vs. on motion of the state vs. on motion of the defendant?

I recently had a case transferred venue, but we had everyone consent that all the usual personnel would go, just have a different jury pool. But it is my understanding that if there is no consent by the defendant to do that, then no one but the defendant and his attorney go - that is new judge, prosecutor, clerk, jurors, etc. (although I am sure the other prosecutor would not mind your help).

There is nothing in 31.09 to suggest that the prosecutor doesn't go along with the case on a change of venue. That would be crazy. Can you imagine how many defendants would be filing motions to change venue if it meant they could also get rid of the mean old prosecutor who is so wrongly persecuting them?

Indeed. If you can't get the prosecutor recused, then change venue and get rid of him/her that way. Venue is just that: the location of the trial (and the resultant jury pool). Nothing else need be changed.