Yoru sentences are very run-on which makes what your trying to say garbled.
You did not link to the pulse generator to answer such things as what is the range of Vcc.
Now, I might expect the value of your sensor to be closer to 5.00 rather than 4.88. It's a little more than 10% low.
Not knowing what you have, I'm hesitant of giving you this information. If you can light a LED with your new found pulse generator, you could be "in like flint".
A LED requires a series resistor and abut 10 mA usually. Now, I'm lost - no information)
Most sensors are either a transistor (commonly called open collector) that closes to ground or a switch that closes to ground. Once you get a LED to pulse, you can replace the LED with an OPTOcoupler such as a 4N26 or something better. The Emitter of the optocoupler would go to ground and the collector would be the output that simulates the sensor.
SO my suggestions here use a separate power supply for the function generator (my crystal ball is broken (MCMisB)). I'm assuming it can source 10 mA ( (MCMisB) etc. etc. Concerns are, we don't have enough info to go on. IF you can operate the function generator on the 4.88 V, then the opto can be elimiated and replaced by a resistor and an NPN transistor like the 2n3904. You could have used a 555 timer or the low power CMOS version, the 7555 timer.
I was hoping to find a schematic for a sensor. e.g. **broken link removed** here it says 5-24V and current sourcing and sinking, The sinking is the NPN transistor.