Don't you need some set 'rules' about all of this?
In advance of the actual appts?
My mum will not have NHS transport for the same reasons- a 20 min journey can take 1.5 hrs all round the houses, so it's a no-no.
If your dad feels the same way, you need discount it altogether.
That leaves a taxi or a lift from family or his friends.
If you book him a taxi and he doesn't want it on the day, you are entitled to refuse to take time off work or whatever.
I think your use of language is very emotive- you say he 'moaned' about hospital transport. Maybe he had good reason? Elderly people, often men with enlarged prostates, can't wait for an hour for a wee if the bus is going all round the houses.
There are probably good reasons why he doesn't want a long journey.
There may be a reason why he is wary of taxis- I don't know. But you seem to look on his behaviour as beligerance, rather than trying to imagine there may be a reason behind it.
It sounds as if there is no joined up thinking behind his care.
Can you not talk to his friends and agree what happens with lifts?
The way my mum works this all is
a) she will take a taxi
b) if possible my sibling (who lives nearby) will offer if they can leave work for a while or do flexi hours
c) if she books a taxi but her friends hear about it, they often offer to take her instead.
Can you contact your dad's friends and tell them that you / he has booked a taxi and they don't need bother unless he's really ill and can't cope with a taxi ride?
Given these are regular outings, it ought to be possible to plan ahead.