Quick advice on remote car starter?

I suggest that you try a new car battery. When the engine cranks, the battery voltage will always drop quite a lot. The electronic parts of the car that are needed to start the car will be made to keep working during cranking, but other stuff like the heating, aircon, and radio systems will often stop.

It sounds like the battery is getting old and during cranking the voltage is now just a bit too low for the remote starter. The OE stuff that responds to the key is managing to work to a lower voltage.

You could fit something to keep the voltage up during cranking. A lot of cars with start/stop systems have a second battery or a battery boost system so that the lights and radio can keep going during cranking.

However, for your car, it would take a lot of work and just fitting a replacement battery will be a lot easier.

If you can fit a larger battery, I suggest you do so. Even a small increase in size of battery can increase the life of the battery a lot. On one car I had, the original battery only lasted a couple of years. The replacement was about 20% longer and was the largest that would fit the battery tray and it lasted 5 years.