How to extend relay working time without controlling signal
I checked out the remote switch description and it is certainly a weird device. They claim it can adjust the brightness, which means it is not just a relay. Something else is going on inside, maybe a diode to cut power in half. So, I would not recommend shorting the input and output of it.

Anyway, I don't think it is possible to do what you want only with parts that you have already. Here is the simplest circuit I could come up with. It requires another timer and a couple of NO relays.

1) Initially NC contacts of Timer 1 provide power to remote switch. Everything else is OFF.
2) When Remote is activated it triggers Relay 1. This relay powers the load and starts both timers.
3) In addition, Relay 2 is triggered via NC contacts of Timer 2 and it immediately bypasses Relay 1 locking itself (and both timers) in ON state.
4) Timer 1 is set to very short delay. After this delay Timer 1 disconnects Remote switch, giving it plenty of time to reset to OFF state.
5) When remote switch fully shuts down it releases Relay 1. However both timers and Relay 2 remain under power provided by Relay 2 bypass.
6) When timer 2 reaches the end of its own delay it disconnects Relay 2. Here is where magic happens - unlike your circuit, timer does not cut it's own power (which causes it to reconnect right back). Instead it cuts power to relay coil, and since both timer and relay are powered by relay itself it is guaranteed that by the time NC contacts reconnect there will be no electricity to conduct.
7) Timers are powered down, Timer 1 restores power to remote switch, and we are back at initial state.

UPDATE:

Here is a photo of wireless remote I found here. It shows the switch controlled by grey slider under the button, which means the remote is capable of transmitting at least 2 different commands. What the receiver does with them is completely different question.



I suggest taking the receiver apart and tracing the connections from the outputs. If they go straight to the inputs with relay interrupting Hot line then you can cut those traces and short them on the relay side. This will give you relay contacts isolated from the inputs so that you can use Remote switch as Relay 1 on the schematics above. Then you'll only need one more relay and one timer.

Of course if there are other parts connected to output traces it would require a bit more tweaking. Also it would not work if the default state of the relay is ON when power is not applied to the remote. You can check this without cutting traces by testing the connection between input and output hot lines.