"convert the entire traditional IRA balance"
This is a taxpayer choice and has nothing to do with a backdoor conversion Roth, so please don't call it that.
taxpayer could recharacterize and then convert $300 from traditional to Roth.
Tax would be on some amount less than $300.
(that still puts you in the situation of having a basis going forward).
The backdoor Roth, when available, is a tax-free event, and leaves no prior years basis.
