Over the last couple years I have had a nightmare with ingrowing and fungal infection in my big toes.
I cannot advise this approach, because I am sure any decent GP or medical professional will tell you absolutely NOT to do what I have done.
After months of tenderness (which was stopping me from climbing more than a few problems, I couldn't even dream of finishing a route without significant pain), swelling and eventually leaking some puss, I bit the bullet.
I took a scalpel and a set of side-cutters and went to work cutting back and cleaning the ingrown and infected parts of the nails on both toes (effectively what the Chiropodist might choose to do). I also went back to applying the anti-fungal stuff you can get from the super market, daily. Since then the situation has improved massively, and every now and again I have had to cut back small sections of the nail which have gotten re-infected. Since doing this cutting and then allowing the nail to grow back, the swelling has stopped and the tenderness that was stopping me from climbing has gone completely. My left nail is back to a normal shape, colour and the in-growing has stopped. The right toe didn't quite clean up as well, so there are still some sections of nail missing, but it looks infection free and is growing back nicely.
What I am trying to say is, don't let it persist because it can get quite bad. Really I should have tried to see a specialist, and you probably should too if it becomes a regular thing. Perhaps your choice of shoe is also perturbing the problem. I found that the pain and swelling was at its worst when I was wearing Scarpa Instinct Lace. Weirdly, having gone back to reasonably tight Dragons (and with all the other work, and the addition of little toe condoms) the problem has more or less gone.
For the sake of your trip, I would recommend at least getting a set of those toe 'protectors' (condoms) they sell at the Depot. These make a big difference, as they really reduce the rubbing and protects the area that gets infected from all the bacteria in your shoes. Like these: https://www.amazon.co.uk/PEDIMENDTM-Silicone-Gel-Protector-1PAIR/dp/B06XG9XHKP
I think this problem has been the absolute biggest limiting factor in my climbing, and I would be climbing significantly harder stuff if I had just got my nails (and athletes foot which set this all off) sorted out when they started to get bad. Let this be a lesson to anyone who might as stupid as I am.