Whilst I understand some of the invective here from staff as regards non-folding cycles, I think that it is unreasonable for folding cycles to attract the same criticism.
I have a Brompton that I occasionally take to London by train, not normally in the peak hours, (too expensive for leisure purposes), but Thameslink trains can get pretty full for much of the off-peak hours. I fold it and it has a cover to prevent any dirt/oil rubbing off anywhere, so as far as I am concerned, it is just another piece of luggage, within the size permitted by the byelaws.
With two airports on the line, there are plenty of inconsiderate passengers with their oversize luggage and not-folded push-chairs big enough for two ten year-olds and their travel accoutrements. I can even legally take it on the tube if its convenient for me. I don't put it across the doors and not between 4-seats so if anybody wishes to have a whinge, (not happened yet), I would suggest that they check the CoC.
As far a charging for it, - well if they charge for all luggage, including (folded) push-chairs, then I would not object.