Will Bundle Builder allow the customer to buy the bundle if some of the variants are out of stock, but set to sell even when out of stock in my store?

Yes. 

Bundle Builder only prevents a bundle from being purchased when some product added to the bundle is out of stock but is prohibited to sell when out of stock in your store. 
If you have the setting "Allow customers to purchase this product when it's out of stock" enabled, Bundle Builder will just subtract the product from inventory and doesn't mind if the number goes negative in this case. 

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