esp_slip_router VS u-boot_mod

Unfortunately there seems to be very little documentation on the PPP mode of the module you got. So I don't know if there are settings to tune in there. I used Zimodem and 8266 Wifi Modem firmware which I would have suggested maybe trying on your hardware except it looks like they don't have PPP support. There doesn't seem to be another ESP8266-based PPP solution that I can find, although there is this that may let you try SLIP. It's not entirely ideal since you lose the rest of the modem functionality but it might be an interesting test at least. Annoyingly there seems to be PPP support for various 8266 firmwares but only a couple that expose it in remotely friendly ways like TON module does.

WifiModem or Serial Line IP setup. I use the SLIP setup on Amigas - but I have also used it on Linux for fun... its not dialup, but it is the real internet.

So perhaps you can consider reflashing your device to use https://github.com/martin-ger/esp_slip_router and then you may be able to get online with a TCP stack running SLIP. My adventures with the TF330 WiFi and being able to get on the internet are described on the AmigaRetro Forum might help step through what you need to do on the amiga side for this (hint it took a little bit of effort to get there and it's very slow going to modern sites but if you wanted to download stuff off your NAS or cruise Aminet its probably okay)