Are you open to the idea of replacing the Fire Stick? There are loads of Android TV boxes you can pick up at all sorts of prices these days. I ditched the Fire Stick and went with a Formuler Z8 Pro a few years ago now, and haven’t looked back (my use case is very specific so a Formuler box might be overkill for your situation). The Turewell T9 got some good reviews and is fairly cheap.
I did that too!
Possibly. Now I’ve exhausted the VPN possibilities for the Fire TV Stick, I may well look at that.
What speed internet do you have?
500Mb fibre - 260Mb measured by the FireTV stick.
Then I am out of suggestions.
Have you tried using the Surfshark extension on the Google Chrome browser? I too had issues with normal Surfshark VPN and was advised by them to instal their extension and now iPlayer, All4 and ITV X work fine 
both in France and Brazil.
Played something on Netflix yesterday, and what still absolutely baffles me is why the firestick manages to play anything from Prime or Netflix seamlessly, but has great trouble with just about every other UK channel.
I don’t think Prime Video UK worked at all for me regardless of VPN. We watch Netflix France - it’s got most of what we would have watched on Netflix UK.
Surfshark worked fine on my phone and PC. It was on the Amazon Fire TV Stick that it was useless. Anyway, I’ve signed up for a 2 year deal with CyberGhost and that’s better than SurfShark on the Fire Stick.
I’m going to try one more thing - there is a video on YouTube from someone in the UK who believes he solved similar issues by investing in a Fire Stick Ethernet adapter, implying it was a WiFi issue that was the problem. And I doubt he was using a VPN at all. I’m dubious, but I’ve ordered the Ethernet adapter. If that still doesn’t fix it, I will look at a Google TV box.
If it’s not your cache - what “quality” settings are you using? Everything defaults to eat your your data in seconds these days. You can change to “lower” quality on the Firestick - you can change iPlayer but can’t remember if that’s the app as well as the website.
I use one of those on my main TV, and WiFi on my 2nd TV.
Sometimes the WiFi connection is betterthan the ethernet!!
Don’t say that! 
Just to recap, I have a 500mb fibre connection and each TV is getting over 200mb (measured by the Fire Stick WiFi tool). The one in the kitchen is a standard 1920x1080 FHD and I have a 4K Fire Stick on the bigger TV in the living room. They both have exactly the same issues, regardless of VPN used. The problem only happens with BBC iPlayer, so I can only surmise there is something dodgy with the Fire Stick iPlayer app - probably in buffer management. Why connecting over Ethernet rather than WiFi should make a difference in my setup is debatable, but I’m going to give it a try - I’m desperate 
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Well, well. The Fire Stick Ethernet adapter arrived today and I’ve just used it for the last hour. The chap on YouTube may have a point. BBC iPlayer hasn’t just randomly buffered and restarted at an earlier point in time like it usually does within a few minutes. I did manage to make it do that by rather aggressively using the pause/rewind/fast forward controls, though. But it’s noticeably more stable. I’ve seen a couple of buffering moments but they were only transient and the stream reliably restarted without any loss.
I’m connecting the ethernet via powerline AV adapters, which is far from ideal. The bandwidth will be significantly reduced. My wife was also busy on a Zoom call upstairs, so this is encouraging, so far…
I’ve just bought a cheap-ish Android TV box. I’ve concluded that FireTV Sticks either don’t have enough resources for the iPlayer to work with a VPN, or it’s deliberately been knobbled.
The Android box I bought is a slightly more powerful one than yours (AMLogic SW905W2 SoC) with 4GB RAM.
The good news is that iPlayer no longer has VPN problems. Yay! (Using Cyberghost)
The bad news is that Android TV with a remote control is pretty awful as you need to frequently switch to a mouse pointer mode. The iPlayer app doesn’t support the remote without mouse the pointer.
Google TV is probably the ultimate solution but the hardware supporting this is a fair bit more expensive.
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For the record…
We use a Firestick & Nord VPN to watch iPlayer, with no issues apart from some occasional minor buffering (greatly reduced now we have fibre).
I’ve been running iPlayer on the Android box on and off all day and it hasn’t buffered once; rock solid.
I did find that of the two Fire TV sticks we have (one HD and one 4K) the 4K one is better, but not perfect. This may be because it has more hardware resources.
NordVPN is quite a bit more expensive than Cyberghost. I did trial ExpressVPN, which has one of the best reputations, and it made little difference, so I concluded it wasn’t the VPN that was the problem.
I have a couple of firesticks, almost gave up on them as they were so slow, or just totally unreactive. But then something somewhere changed and both work wonderfully………touching wood as I write! And I also use them with nordvpn 