"Elements of Buddhism can be rationally accepted such as the suffering that arises through attachment, the benefits of meditation, and even the acceptance of anatta, or non-self. It seems, however, that a belief in rebirth cannot be accepted rationally."
Indeed, the Buddha described his Dhamma as "The Dhamma visible here & now". However, if that visibility were immediately accessible, we wouldn't have doubts about it, and it would not be necessary a "path" to help us "see it". So, knowledge and skills are necessary to develop understanding, just like knowledge and skills are necessary in sciences to "see" phenomena (e.g. whether the claim of climate change affected by humans is true or not).
In the same way, knowledge and skills are necessary to see past lives according to the Buddha.
"When my concentrated mind was thus purified, bright, unblemished, rid of imperfection, malleable, wieldy, steady, and attained to imperturbability, I directed it to knowledge of the recollection of past abodes. I recollected my manifold past abodes, that is, one birth, two births ... 'There I was so named, of such a clan, with such an appearance' ..."
-- MN 36
"A believer must suspend rationality to accept a theory that describes the migration of some level of consciousness across a lifetime."
One does not need to accept (believe) or reject a theory that one is unable to verify at the moment. One can simply be aware of the competing theories and strive to see which one, if any of them, is true.
In science something similar happens where researchers point out flaws or remaining questions that need to be answer (i.e. that need more research) before accepting a theory.
I find this similar to the way the Buddha taught as well (e.g. refer to the discourse of DN 1). Thus, generally speaking, it's better to try out methods to see the truth than to hold or cling to any view prematurely. Or, more than that:
Who is attached still enters into doctrinal debates,
but one unattached, how could he take sides?
For him nothing is taken up or put down,
With all views shaken off, relying on none.
-- Snp 4.3
Finally, the idea of "what transmigrates" is an old and on going debate, as the Buddha explicitly denied that any substance transmigrates, which would make that very substance something that endures -- that is, a Self (see this question: Then where did the concept of “rebirth” come from?).
"A common analogy used is the light emitting from a candle. Nothing substantial transmigrates, however something passes over from one life to the next. This theory cannot be directly perceived. Therefore, it cannot be tested, or verified rationally. It is simply a metaphysical inference."
As explained above, the buddhist doctrine asserts this knowledge can be directly perceived without resorting to fragile or indirect means (like beliefs). So, in the context of buddhism, recalling past lives is as metaphysical and irrational as the idea of remembering what one did a few moments ago.
"Buddhism encourages debate and questioning of its theories, and to not accept everything on face value. How then can a Buddhist test this theory"
I'm not able to recall my past lives, so my knowledge is not rooted in practice, only in reading. The progression of meditative attainments the texts shows seem to indicate that one who has mastered the fourth jhana (samadhi practice) possess enough skills in concentration to direct his/her mind to memories of past lives. I've heard that one practice is to gradually recall childhood, infancy, even memories in the womb, and then, to the past life, and so on.
"Can one disregard this element of Buddhism? Is it an essential aspect of the philosophy?"
This question has been asked here: Is rebirth/reincarnation central to Buddhism?.
There is, though, a Secular Buddhism movement in the west which seem to, literaly, strip out a lot of things from Buddhism, including rebirth. Also, some buddhists who may not subscribe to this movement may also believe rebirth (as in dying and reappearing in another life) is not true or not part of what the Buddha taught.