I suspect that Bercow's outburst was more to do with the Brexit negotiations than any liberal sentiments that he might hold.
Making sure that the UK is not viewed positively by President Trump would make a good trade deal less likely and would give the careerist remoaners in parliament more leverage.

As leader of the UK government Mrs May should have reserved the right to carry out Brexit without interference (she has the mandate of the EU referendum result) and as such can claim to speak for the British people.

The government team are dealing with unscrupulous individuals who are at best unwilling to accept the democratic decision of the UK electorate and at worst, willing to sabotage the Exit negotiations to protect a severely flawed ideology as well as a corrupt and dysfunctional Union.