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That ebay amp can drive 50 ohms with a whopping 10dBm, or 10 milliwatts.
That works out to about 0.707 volts RMS on the output, less than your Si5351.
It's meant for use in front of a receiver, or something like that.

A comparator that would work well at HF with a 10 volt swing might not be cheap.

Do you have QRP transmitter lying about capable of at least 2 Watts?
Drive that QRP transmitter with the Si5351,
put a 50 ohm dummy load on the QRP transmitter so you don't blow it up,
and run coax from that dummy load into your vacuum tube transmitter.

A two watt transmitter into 50 ohms is? squareroot(2*50) = 10 volts RMS?
If it's more than two watts, you could bring it down by a special resistive divider dummy load.

Might work.
Or it might trash the QRP transmitter if you do it wrong.
Wouldn't be a bad idea to seek some help from somebody with experience
in this sort of thing and some test equipment.?

Jerry