Here is a comparison of the two types of comparator outputs:
Everything inside the gray rectangles is inside the comparator. The push-pull type has an internal connection to V+. It can source or sink current. The open-collector (also called, open-drain) type can only sink current to ground when the output transistor is turned on. An external resistor ("pull-up" resistor), makes the output voltage high when the output transistor is off. Current is sourced through that resistor. For more detail see:
Comparator - Wikipedia, the free encyclopedia. Scroll to Output type.
Can your LM393 replace the TVL3702? I don't know for sure, but you can give it a try. The circuit does not appear to be too demanding on the comparators, except they need to source current. So, you could try a pull-up resistor on each output.
What size resistor? For the greatest voltage swing, you would use a large resistor, such as 10K; however, that won't be able to provide much current. On the other hand, a much smaller resistor will use quite a bit of current when the output transistor turns on, because it will be connected directly from V+ to ground minus the voltage drop of that output transistor. You want both good current and a large voltage swing to turn on the 2N7000 mosfet. I would try a 1K resistor (the LM393 is limited to 18 mA current sink on its output). If that doesn't turn on the mosfet, then I would try 4.7K or whatever resistor you have that is larger.
Please give us updates. There are numerous push-pull comparators made. You may be able to get one of them instead of the TLV3702, if these modifications do not work.
John