433Mhz Receiver Low Range Only on Raspberry Pi

Your decision to use a capacitor was the correct idea, but that one is much too small.

It's unclear what model you have, but your RPi probably consumes between 700-1000mA. However, if you calculate that 2200uF capacitor's discharge curve for 5V @ 700mA, it's only going to delay a brownout for maybe a millisecond before it drops below the RPi's minimum 4.75V. And that's assuming your battery is brand-new and fully charged.

That RS-540 motor can draw a massive amount of current for a few seconds when starting or stalled, and that capacitor will be discharged long before it's over. Some motor controllers I've seen designed for the RS-540 are rated to handle 30A for 5 seconds.

Another option is to use a battery with a larger mAh (in addition to a larger capacitor) that can handle higher current draws.

You may still want to consider a second battery so the RPi can remain functional even after the motor battery has been completely exhausted.

I have a reference design for a similar type of 433MHz transmitter / receiver modules and while I can't share the schematic due to an NDA for this particular pair of modules connecting the transmitter output directly to the receiver could exceed the absolute maximum emitter to base voltage on the input stage of the receiver. It would depend on what particular transistor your module uses and its configuration but I wouldn't consider it a safe thing to do in the absense of other information and often the transistors aren't clearly marked.

I've never actually tried doing this but looking at the receiver circuit using a low-side switch on the path to ground would prevent that and might be worth a try, although it looks like it may end up radiating some RF back into the positive supply rail so you might need some extra filtering from the positive supply rail on the receiver to ground.

Depending on your microcontroller (if using one) and its current sinking capability you could try switching the ground of the receiver using a floating / low I/O pin, although it's hard to know how well the mix of RF and digital circuitry would work together. Otherwise a fairly neat alternative might be to use something like figure 8 speaker cable with a different conductor for the transmitter and receiver.