The shooting orgs are in a really difficult spot with this one. It's quite clear that FLDs were not happy that the 'self certification' process was reliable and were determined to do something about it by requiring GP confirmation that it is correct.
In some ways, the fact it isn't in the HO guidance makes the situation worse, it means FLDs make their own rules within their constabulary. At least if it were dictated from the top down, there would be some consistency. It would also make lobbying easier, because you just have to convince the HO and not 90% of Chief Constables. Given this is now so entrenched, a national approach seems the only logical outcome, with set procedures to be followed and fixed charges, so as to ensure no-one is taken advantage of or refused a renewal because of the current approach.
Back to the point, if the orgs opposed it completely they would risk appearing to not care about public safety. If they succeeded in opposition and someone did go on to kill in a situation where the perpetrator lied about their conditions, the blowback on the shooting community would be considerable, as the Police would wash their hands of responsibility and say 'told you so'.
Now it's here, other problems arise as noted, GPs conscientiously object or charge fees out of proportion to the time commitment required, either profiteering or (more likely in my view) simply because they don't want to do it.
The orgs could make a lot of noise about this but, again, where does that take things. The NHS contract is such as to mean there is no mechanism to force GPs and, in the court of public opinion, I'm not sure I'd back shooters over doctors. Conscientious objection is a different matter, most doctors I know love to go on about how non-judgmental they are of patients' life choices. For some reason there seems to be a caveat for shooting.
As to who pays, the orgs could push for the Police to meet the charge. However, forces are already complaining that the real cost of administering licensing is out of all proportion to the fee charged. To argue the police should meet the GP fee would open up a larger debate about the overall. It doesn't seem that implausible that they might get the GP fee paid by the police only to have the application fee varied to the true cost, so an overall loss for shooters.
I fear medical verification is here to stay. While I'm not convinced of Connor's suggestion that we might want the Firearms Act totally overhauled (not because it isn't a good idea in theory but because so commonly you end up with a worse mess than the previous) I do think there needs to be a uniform approach underlined by a proper system for providing the verification, if FLDs are to continue to require this.