JT - that 3 year criteria is a nonsense. Who's to say how long you have been 'trying' for unless there is some big brother monitoring system in your bedroom checking for sexual activity and/or condom use.
I didn't mention this earlier as I don't want to give you false hope but I am 41 now, treated by the NHS at 40 after one failed nhs full cycle, one failed private full cycle and one failed NHS frozen transfer. You could have knocked me down with a feather when I got a call offering a final nhs full cycle at 40. I was told this was as there had been an unreasonable delay in treating me when I started the process. A couple of years back I did indeed make a complaint about some of my treatment but I had no idea they would offer me this. I feel so lucky and the prejudice against nhs funded IVF angers me a lot.
One thing you could do, and I did before my final cycle, is take DHEA. This is a non-licensed (but not illegal) drug that you need to order from the states (google micronised DHEA). I took this on the recommendation of both my private and NHS consultant. In fact, although I ordered my own, the private consultant had a stash in his filing cabinet. There is some evidence to suggest that this helps conception in older women with low AMH.
I just looked up my AMH - it was 4.27 so close to yours. I always got eggs on a cycle. Don't be discouraged by tales of women who get 18-19 etc - often a good proportion of these are immature and won;t fertilise or develop well. When the NHS offered the final (successful) cycle they did ask me to do another AMH to check it hadn't dropped further. It was then 5.03 so not significantly higher but they were happy to go ahead.
I think what kept me sane in all of this was always having a b plan. My final, on the last cycle, b plan was to go for donor treatment in Cyprus (v. high success rate, anonymity, instant availability and a good choice of matches as my colouring is dark). We were also scoping out adoption routes too. Obviously this may not be for everyone but it's worth thinking more broadly than just the nhs treatment so there is always a chink of light around the corner for you.
Good luck