Blackjack "Doubling for Less"

There is one casino (that I know) that allows a "double for less" after you have already doubled once in place of a full double-double or even a double-double-double. Your "double for less" can be for $1, whether your initial wager is $3 or $300. However, if you chose this option, you may not take other cards, including doubling again.
The game: Spanish 21, H17, 5 decks, Redoubling (three times for up to eight times your original bet), splitting any card - and re-splitting to up to four hands, doubling after splitting

Typical scenario: You have placed a wager of *whatever* and are dealt 7-4 against 8. You double and draw "A." In basic strategy for this game the correct play is to re-double (assuming TC -4). However here it seems (to me anyhow) that tossing $1 out as a "double for less" seems to make a lot of sense, instead of the full double-double. It's less clear, however, if you have doubled to a stiff hand against a bust card (where you typically stay) or 9, X, A (where you typically forfeit).

I am wondering if there has been a good analysis of the best way to utilize this option? Clearly the usefulness of this tool would depend on the count. Moreover, an extra $1 on top of a $6 (after doubling) bet is very different statistically than $1 after a $600 (after doubling) bet. I have some ideas how to approach this, but don't have the ability to run the simulations to test them. I've done some searching online and have not found much. It seems to be a bit of an odd rule, but has got to have some utility that I can exploit effectively. I'd love to hear what those of you with more experience than me (which is almost all of you) think on this.

Thanks in advance.

Source.

 

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