There is a relatively simple algorithm, depending on what you think is relatively simple. We'll just need to assign some numbers to the chromatic scale, with the major third in the key being 1 and the minor third being -1. You can extend this as far as you need to, so:
Now just change the sign. I in this key (CEG) is -4 1 4. Changing signs yields 4 -1 -4, or G Eb C, just as expected. V7 (4 8 -2 2) becomes -4 -8 2 -2, or C G# F D: the inversion of iiø, as expected. It doesn't matter if you use 8 for B or -5.
This might seem a little clunky at first, but it's not that much different from on the fly transposition, which is fairly simple to learn how to do. You could also think about it in scale degrees, instead of trying to get absolute names, and then you would only have to work it once, as long as you're comfortable working with relative degree.
In this chart, you'll see the note transformation, but I've also put the new scale degree in, and you'll notice that the root becomes the 5th, the 2nd becomes the 4th, &c in inverse proportion to the scale degree, so in practice, you can count down from the 5th as your original note goes up. In this case let's take V7, so scale degrees 5 7 2 4. That means count down a 5th from G (if in C), a 7th from the G, 2nd from the G and 4th from G, giving us C, Ab, F, and D, the expected iiø.
This also means, if you are familiar with interval inversion, that you can calculate the negative counterpart from the inversion. Perform the interval inversion (root = root, 2 = b7, 3 = b6, 4 = 5, etc) and then go down a fourth from that, and that will be the negative degree.
So to put this to practice, if your chord contains the root, the 3rd and the 5th, then the inversions are root, b6 and 4. Go down a fourth from those and you have 5, b3 and root, or G, Eb & C, as expected. Any one of these methods should work, will not require pre-charting, and should be able to be performed on the fly once you are used to it.
