well, the selection of a microcontroller is a very personal thing. It even varies project by project... the right tool to do the right job.
However, in your case, it would be more of an availability issue. Which ones can you buy? Which ones can you program? If you can't now, how much can you afford to buy one???
I use the AVR, ATMEGA series. You can be up and programming them for less than $50. All you would have to buy is a programmer ($29+tax/shipping at digikey.com) and then download the development software for free Atmel. I use Studio4. I know it is older, but I do all my programming in assembly and I'm used to it (you don't want to know how many years it took me to upgrade from studio3)...
Probably too big for what you want to do. You can probably use an 8 pin micro that has upto 6 I/O would be all you need.
You won't need a micro for the transmit side, just the encoder hooked up to the transmitter module. For the receiver side, the two outputs (or more if you have the I/O, why limit) that come out of the decoder feed two port pins on the micro. The micro can then drive a transistor to turn on a relay to give you an output. The micro can do all the timing. You never did say how long of a delay you needed... I see three modes, momentary (on while receiving signal), timed (signal starts a timer), and latched (on with signal, then off with the next signal, etc.), or you can use one channel to turn on a latched relay, and a second channel to turn off the relay, eliminating the need for a micro.
on the data sheet I posted, it shows the connection. The encoder just needs power and the pushbuttons. It then goes to the tranmsit module. This module just has a signal input, an enable pin which you would tie to always enable, power, and an output to an antenna.
On the schematic posted this post, the 750 ohm R10 can be lowered to 300 ohm by using 3.3V. It limits the power output so you don't get into trouble from the FCC. The data out of the encoder would be connected to the data pin (2) of the transmit module. Don't let PDL going to RxD fool you. it is NOT bidirectional. I use that pin as an output to enable/disable the transmitter.
The receive module is the same, only in reverse.
Don't worry about the RSSI signal, that's just receive signal strength. I use it to help filter my results.