That’s not quite correct mate.
If you’re convicted in the magistrates court (as Barton was) then you have automatic right of appeal to the crown court against conviction and/or sentence.
If you are appealing the latter then the crown court has the power to increase the sentence, but only up to the maximum that the magistrates court had the power to impose (which is six months imprisonment for this offence).
So if he appealed his sentence then that would be at large.
In reality that power is rarely exercised because it is custom in cases where the judge/recorder presiding is minded to increase sentence (either at the outset if it’s just an appeal against sentence, or after the appeal against conviction is dismissed) to ask the appellant’s legal representative if their client is ‘aware of the court’s powers on sentence’. That is code for ‘abandon the appeal against sentence now, or we’re increasing it’! Best not to ignore that warning!
I expect Barton will appeal the conviction and not the sentence, and he may as well. The test for an appeal to the crown court from the magistrates is far less exacting than one from the crown court to the court of appeal, where the test is that the conviction is ‘unsafe’. In the former it’s a rehearing which means he gets another bite at the cherry, so he may as well go for it, especially as he’s had a dress rehearsal previously.