From the BBC piece:
The families are paying for the private aspects of the wedding.
However, republican campaigners say taxpayers face an estimated £2m security bill.
The campaign group Republic argues that as Princess Eugenie, the ninth in line to the throne, does not carry out public duties, the families should foot the bill.
Buckingham Palace says this is a matter for the police.
I wonder whether the 'republican campaigners' have explained how the 2 million figure was arrived at. Often these 'costings' from campaign groups mix up fixed and variable costs. The figure, if broadly true, does seem high but if it is because the senior royals are attending then i suppose there would be security costs (but surely not anything like £2m) if they were opening a hospital or attending a festival etc, so it's just the price we pay for having senior royals.
Anyway, the "I can’t see what qualifies her to have such a public televised wedding" part of the OP seems to have been cleared up. The only 'qualification' is that a commercial channel (its advertisers) feels it's worth paying for