I think this is a very interesting question, and one I think about a lot, but there’s a few major issues with perspective that I think lead to a lot of answers that don’t really make a whole lot of sense.
For one, as was pointed out up thread, what exactly does random mean? I might imagine it means that if we could create a list of every possible physical constant and created a universe for every possible combination, then our universe is just one of a likely infinite number of other possibilities, with likely many other universes not all to different from ours also containing life. But sort of randomness begs the question of no intelligent design so, of course, any deductions from that will favor that assumption. Similarly, if we suppose that only a single universe exists with these precise values of those constants, then it has the opposite affect and tends toward making the universe appear to be intelligently designed.
Second, we really don’t know a whole lot about how all of these physical constants interact, nor do we understand a large number of other aspects of what makes our universe tick. It may in fact be that only a very narrow band of these constants is actually consistent in anyway with anything existing, not just with various matter and energy distributions.
Third, and perhaps most importantly, using any current knowledge or observations as evidence against design are flat out ridiculous. For instance, imagine we were to visit Earth some billions of years in the past and observe the earlist microbes. Without knowledge of how it would develop, we’d probably see most people argue that the life is extremely primitive and the probability of it developing into anything resembling us would be so vanishingly small as to be irrelevant, and they’d be accurate in those chances but ultimately wrong. And so to argue the same about a relatively young universe, the scarcity of life, and the seemingly under-developing technology and all that is to lose perspective on the true vastness of time and space of the universe. In a random universe, over such vast time scales, we’d probably expect life to randomly appear and reach various states of development countless times over the lifetime of the universe. In a designed universe, we might expect life to form and continue to grow and spread, but in either case, we have no meaningful way from our current vantage point to make any sort of judgments about either of those possibilities or other sort of combinations. It’s like listening to a piece of music and attempting to determine if it was intelligently composed or just a series of random notes, but basing that judgment off of the first brief moments rather than off of any amount of substance of the piece as a whole.
In short, I don’t think there’s any meaningful way to differentiate a designed vs. undesigned universe because it requires such a large number of unverifiable assumptions and unobservable observations that I don’t think we can explore those possibilities meaningfully in that way. In fact, even as a theist myself, I think there’s likely no difference and that that is wholely consistent with both possibilities. Obviously, if it is random, then anything that is possible has some probability of occuring. And of a designed universe, any being intelligent and powerful enough to create it could and would so from initial state and, thus, any of his interactions with the universe would be natural. As such, these questions make the most sense, I think, explored in a realm other than science.