When we did Beachy Head Cave we were advised by the Coastguard at Birling Gap to phone the coastguard's south coast control centre to avoid such a callout.
They can be contacted on 023 9255 2100 and were most courteous, thanking me for notifying them and asking that we called them back when we had finished.
This is actually quite a good idea for Beachy Head Cave as you are right opposite the lighthouse which (when I was there) is manned, they are quite likely to spot you and possibly call out the coastguard.
Also you have a built in callout system which ensures that the coastguard are the first to know about the cave rescue (like most cave rescues
)
The other advice the Coastguard at Birling Gap gave us was to approach th cave from Eastbourne. This is a BAD idea. You have no idea how many clift falls there are to cross and when you get to Beachy Head you look towards Birling Gap longingly, knowing that it is the wrong way and you have to traverse all the falls again to get back to the car.