Transitive swear and Concessive Subjunctives
I don’t know why you think swear is intransitive; it is obviously used transitively here since it is love that would be sworn. If she can swear love, then swear has to be transitive: it has an object.
But had you checked a dictionary, you would have found numerous transitive senses for swear. Indeed, the OED gives no fewer than a dozen transitive senses and subsenses for this verb. For just as you can swear fidelity, so too can you also swear love.
Now that we have that out of the way, what remains is nothing fancier than commonplace sentences like these:
Do it and you’re a dead man.
Give it up and we’ll go home.
Stop over and we’ll talk about it.
Be kind to strangers and strangers will be kind in return.
This type of construction includes “be but sworn my love”. If Juliet swears his love, she attests to it as by promise or oath — upon which circumstance he pledges to renounce his own family.
Technically speaking, this is a concessive subjunctive construction, the same thing we find in frozen refrains and familiar aphorisms nestled in sentences like these:
Be it ever so humble, there’s no place like home.
Give him an inch and he’ll take a mile.
Come rain or come shine, the cows will still need a milking.
Take what you will of me, I will never complain.
As you see, the verb in concessive subjunctive clauses likes these takes a simple infinitive as far as its morphological inflection (or lack thereof) goes.
Shakespeare uses concessive subjunctive constructions quite frequently throughout his writings. In the selfsame play, we later in Act 3 Scene 1 find the following famous example from Mercutio perishing by the sword:
No, ’tis not so deep as a well, nor so wide as a
church-door; but ’tis enough, ’twill serve: ask for
me to-morrow, and you shall find me a grave man.
Although Latin teachers go to some effort explaining Latin subjunctive inflections in concessive clauses, even in uninflected English this sort of thing still happens — at least as far as having concessive clauses goes.
These are really nothing fancy, as the first set of examples in the earlier section prove. Concessive clauses still happen in regular English outside of fossilized phrases like “Be that as it may”, and when they do, they take a bare infinitive in English.
Last and probably least, I believe I have by now also answered by way of earlier demonstration your remaining question as to but’s function in the love-sworn citation.