Tricky issue here but I’d suggest a downward expander rather than a gate.
Put the preview window on to firstly try the downward expander in the dynamics option. Use amplitude statistics to scan the audio where you’re not talking and this will give you an idea on setting the threshold. Adjust the ratio to your personal preference.
If that doesn’t work to your liking you may find a downward expander created yourself using the dynamics processing option may work better as you can set that manually then fine adjust the attack and release times in the settings tab.
A downward expander will often provide a more pleasing audio experience than a gate but with careful use of release times you may achieve what your after using the noise gate in dynamics after all. I’d personally go for the expander as a gate is either open or closed where as using an expander you can set the same threshold AND control how if the sound you still let through with the ratio option that you don’t have with the auto gate option.
You may struggle though to achieve what you want automatically as if the mic bleed is louder than background ambient noise you want to keep your background noise will get either gated or reduced by the downward expander which affects all audio beneath the target threshold set which might then end up sounding unnatural.
The speech volume leveller is a great tool but you’ll have to have the gate or expander set correctly in your processing chain otherwise the speech volume leveller will suck up what you don’t want. Remember you can play with the levelling amount with the preview window open and using your ears also to get around this and the speech volume leveller does also have a noise gate in the drop down options but this may also wipe out the ambient noise you want to keep.
Another option for future recordings is to possibly make yourself a DIY isolator to go round the mics you use. Get a thick bit of card and make equidistant cuts around the card (do not cut all the way through). Bend the card to your desired shape and fill the cuts with glue to hold it in place. Then get some foam and glue this to the inner side of the card. You can paint the card and cover the foam with material if you’d like to look better. Don’t go too big though.
If you can get that around the mics you use to record it may trap out some (unlikely all) of the microphone bleed you’re talking of. Test this carefully before use though as having it too close to the mic may result in “muddy” audio and may produce worse results than you currently have.
Best of luck. Hopefully the expander might do what you want and with the preview window on you’ll be able to get a visual idea of the impact your gating/expanding is having.