Here is our experience May 2023, when we stayed 4 nights at Belle Tout B&B on the cliffs near Beachy Head...
1) Taxis - Of the 3 companies listed on the National Rail page, we had luck with Sussex Cars and no luck with 720 Taxi. The 720 Taxi required prepayment in order to come out to Belle Tout but their text-based prepayment link wouldn't send to a US mobile phone and they would not take my credit card information over the telephone. Sussex Cars didn't require prepayment and we were happy with them (multiple times). My memory is that Sussex Cars only accepted cash, but I might be wrong on that.
2) Another transportation option depending on what time of year/what day of week you are visiting: we used the Eastbourne Sightseeing hop-on hop-off bus to get around and to see a bit of the area (https://www.buses.co.uk/ess/eastbournesightseeing). It's a one-way route, but stops at both the Birling Gap NT (stop #9) and the Tiger Inn in East Dean (stop #11).
3) East Dean has a nice village market each Wednesday: http://www.eastdeanvillage.org.uk/village-market.html If you go, look for Paul and Ronnie who sell their handcrafted chocolate bars in both straightforward and unusual flavors (https://gatoschocolate.co.uk). Try their Caramay-B caramelized white chocolate--I can still remember how good it was.
4) Birling Gap vs Cuckmere Haven: Although Cuckmere Haven gives the "classic" view of the Seven Sisters with the Coastguard cottages in the foreground, we gave that a miss in favor of Birling Gap. At Birling Gap, there is a National Trust (NT) gift shop, small cafe, toilets, nice views, and stairs down to the shingle beach.