Is there a similar word to 'twain' but meaning 'three' instead of 'two'?

As far as meaning goes, you can use "cleave in three" or "cleave in thirds". More obscure and inexact is "cleave in terciles" or tertiles. The latter two words are statistical terms referring to "three [ordered] parts, each containing a third of the population" and "any one of the three groups so divided".

The rationale for "cleave in three" and "cleave in thirds" is that twain means two. In its etymology we find it " survived as a secondary form of two" in various cases, and also "in oral use where it is necessary to be clear that two and not to or too is meant."

Regarding other suggestions: My understanding of the many senses of tierce is that while all of them are related to the number three, none of them would make any sense in a phrase like "*cleave in tierce". Trey means "a playing card with the rank of three" or "a score of three in cards, dice, or dominoes".

I'll add one more, to the same pattern: trine.

I'm familiar with it mostly in astrology, where it refers to a separation of 120 degrees; a grand trine consists of three planets, each in a different sign, all with separations of 120°, forming an equilateral triangle when linked.

And I think cleave in trine would work nicely.

You could try the verb third. It's really no different from 'halve', but it has something of the archaic flavour you seem to be looking for.

(It goes back to Shakespeare: Two Noble Kinsmen i. ii. 96 What man Thirds his owne worth.)

Trifurcate. (a bit too equal?) Trisect. (a bit too accurate?) Tripart. Trichotomize. The OED fizzles out after this.