Buying tow bar/base plate for towing, need anything else?

The tow bar physically connects the car to the RV but the brake lights are a separate issue.
The easiest solution is a set of tow lights, usually magnetic stick-on, that plug into your trailer light connector by the hitch.
I'm assuming your Toyota has a manual transmission. If it's an automatic you don't want to flat-tow it on a tow bar.

No it is an automatic. Can you only flat tow cars that are manual?

I don't know what year your Corolla is but this is from the 2015 Owners Manual Pg 158:

Dinghy towing (except vehicles with a manual transmission)

Your vehicle is not designed to be dinghy towed (with 4 wheels on the ground) behind a motor home.

NOTICE
To avoid serious damage to your vehicle Do not tow your vehicle with four wheels on the ground.

Moved to Dinghy Towing forum from Towing.

It sounds like you need a tow dolly rather than a tow bar.

I tow my Corolla on a Demco dolly as they cannot be flat towed as they come from the factory. You can have a pump installed on the transmission so it could be flat towed. I am not sure how that would affect your warranty. JMO

You need to tow dolly for that setup. But you could have a lube pump installed for the auto tranny. The problem with those is that some times they fail. And when they do you won't know it until the damage has already been done. I'd go with the dolly. And at some point you might want to replace the car with something else, your good to go.

The problem is you will be spinning the parts in your transmission (even if it's in neutral) but the transmission's oil pump is driven from the engine, if it's not running the trans has no oil circulation.
You'll fry the transmission.
A couple times in my life I was in situations where I needed to tow something with the drive wheels on the ground, so I simply started the engine and towed it with it idling.

You need a trailer or a dolly, to get the car's drive wheels OFF the ground. Or you need to buy a tow-car with a manual transmission.