Intuition would lead you to thinking that you need a thicker string to get a similar end result, because you need to compensate for the lower tension caused by the short scale.
In practice, the thicker the string in relation to the scale, the less sustain, note definition and "purity" (just think, playing higher up on the fretboard yields a darker, less articulate tone compared to the first few frets, which can often be used to your advantage by the way; and this gets more pronounced with thicker strings). If increasing the string diameter were all that it's needed, you could get a guitar scale to sound like a bass, just with very thick strings. In practice, it doesn't work that way.
Why do you need string tension in the first place? It has more to do with dynamic (transient) response and feel than with tone per se. Why do heavy genre players generally favour thicker strings? Heavier genres may have faster, denser lines, and may benefit from more overtones, while not having, generally, that much use for sustain. If you're going to pound that E string with 8th or 16th notes at high tempos, you want a string that is nice and stiff and responds with lots of attack and overtones at the transient.
Also, have a look at what guitar players, especially acoustic guitar players do. For strumming, thicker strings are usually favoured because they give brighter, richer sound. But for lots of fingerpicking and solo playing people usually prefer light gauges, and not only for the reduced tension (feel) but also for the increased sustain and "purer" tone (sound).
In my dabblings, I have found that thinner strings actually sound best for warmth and roundness on bass, provide better sustain, and on a short-scale especially, fewer bad overtones and more clarity. In general, if you have a bass with bad overtones at the transient (or if you have a style that favours slower, more melodic lines and you wish for smoother sound) try reducing string thickness. For the "modern vintage" styles that are quite popular nowadays, don't just go for .45-.105s. Try .40-.100s, or even a thinner E string. Or mix and match (thicker treble strings and thinner bass strings). My Danelectro long scale really came to life with Rotosound Funkmaster .30-.95s (even though .35-.95s are a better choice, because the treble strings on the Funkmaster set are just a bit too soft).