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Barlow lenses are curious creatures. Like EPs, some work better in one scope design than they do in others. I've got two, one a humble 1.25" achromatic, the other a 2" ED hot shot, both 2X. They each perform very differently with different scopes I've got and then also with the different EPs I've got. For me, I don't believe they maintain the same viewing properties of an EP as when the EP is used neat. They can't. They alter the shape of the light cone entering the EP, so it will mean that the EP will react differently. For example, my 30mm Superview with the 2" 2X barlow is not a good combination in my fast dobs, giving a worse quality image than the neat Superview, which is a reasonable image. But, the same Barlow with the 5mm Hyperion in the same scopes is a cracker.
I think your Barlow though will do well with the 9mm TMB. Mind you, you will be pushing the scope to approx. 333X, which for really clear, sharp image you will be limited to only a few times a year. When those occasions do happen, you will be happy you've got it up your sleeve. When you first try this combination, and the image is shimmering like a looking through hot water, don't blame the Barlow or EP, its the thermal currents in the atmosphere your scope is picking up. You'll need to wait for that occasion when the atmosphere is more cooperative for high magnification to make the final call on the EP and Barlow combination.
Ultimately, only trial and error will tell you what works with what.
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