TL; DR Same problem here - my 2018 LC with aftermarket remote start shuts off the engine when you approach and open any door. I was /told/ that it's a feature / function of the Toyota electronics / security that remote starters cannot work around - there is no "it just keeps running" option. I hardly believe anything I'm /told/ any more, but I didn't have the energy to run this to ground, so I've just grumbled and lived with it.
Backstory: I had the dealership install a remote start on my CPO 2018, at their cost, as the sole concession on the deal. I thought I was getting the Toyota one, but didn't specify that in any of the paperwork, and in the end got an undetermined aftermarket version installed by an unknown stereo installer business in another town that was subbed out the work by the selling dealership.
The explanation from the dealership was that the Toyota branded remote start is "not that good", principally in terms of range at which it will actuate. I received an authentic "laugh out loud" to my face when I asked if my new aftermarket remote starter would still work with the Toyota smartphone app to let me remote start the LC. I was /told/ that not even the OEM Toyota remote start, when installed at the factory, would allow app-based actuation. Again, I just let it go.
If any of that is true, I can't imagine how bad the Toyota OEM one would be, because this one is pretty bad on its own. There is no separate remote to operate the remote start feature - it requires three "lock" presses on the OEM key to initiate the start. The range is pitiful, worse than any Ford, M-B, BMW or Porsche I've had in the last 10+ years. If there's an intervening vehicle between you and the LC, the probability of a successful start drops by half.