How do I switch a 230V circulation pump transmitter from bathroom lights?

Hello, I have 2 bathrooms whose lighting is connected to one fuse. I would like, at the moment of closing the circuit (lighting up in either bathroom), to switch on a transmitter (I already have one - it is enough to supply 230V) which will start the circulating pump.
I have tried a 230VAC relay, inserting the coil contacts between the fuse and the wires to the bathrooms, and switching to the transmitter.
This solution works half-heartedly - in fact when I switch the light switch the transmitter gets its 230V to start the pump, but the drop in voltage on the coil is so great that the light in the bathroom does not come on.
How can I improve this? Maybe the whole concept doesn't add up?

The challenge in general is: there is current in the bathroom circuit -> feed power to the controller.

http://www.fif.com.pl/produkt/1144/1145

Thank you for your reply. I feel the solution is close at hand.

However, there may be a problem with the device from the link. The minimum switching current is 0.6A or ~138W, and the lighting in the bathrooms is all LEDs. I will still measure how much I actually have in the circuit, but I feel something that the current is unlikely to reach this value and therefore will not switch on the transmitter.

Is there a version that switches on at any current (or 0.01A)?

This relay has a Φ4mm measuring channel, so you can thread the wire/line with the circuit being measured about 3(X) times - then X times the actual measuring current....