esp_slip_router VS Tasmota

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)

All of my office devices are connected to Sonoff S31 smart plugs, flashed with Tasmota on their own locked-down VLAN, and can be controlled via voice, Alexa, Tasker automation on my Android, NFC tags at my office door and various scripts triggered via my StreamDeck (on Linux). This all works fantastic, and I can turn on or off my devices, including my printer, when I start or end my day at work.

The best alternative firmware example for true IOT devices is Tasmota [1]. Erase manufacturer firmware for every ESP devices the day after purchase to avoid those careless manufacturer firmwares.

[1] https://tasmota.github.io/docs/

The ESP8266 based Sonoff S31 and ESP32 based Switchbot plugs, both flashed with either Tasmota or ESPHome, are my go-to options for US WiFi smart plugs with power monitoring.

And if you want to turn your ESP8266 into a standalone appliance, with a web interface and directly editable config that doesn't need a build step, Tasmota is another great option.

https://tasmota.github.io/docs/