From the Kantian perspective, what would be the relationship between our intuitions of space and time (which form the structure of subjective experience and are not things that exist outside of human cognition) and the mathematical models of space and time that are used in theories of physics? Is Kant going to deny that these mathematical models have mind-independent existence and instead say that they are intelligible only through being abstracted from our intuitions of space and time? Do these mathematical models of space and time pose a problem for Kant somehow?
Mathematical models of space and time doesn't pose a problem for Kant according to reference here:
In 1781, Immanuel Kant published the Critique of Pure Reason, one of the most influential works in the history of the philosophy of space and time. He describes time as an a priori notion that, together with other a priori notions such as space, allows us to comprehend sense experience. Kant holds that neither space nor time are substance, entities in themselves, or learned by experience; he holds, rather, that both are elements of a systematic framework we use to structure our experience. Spatial measurements are used to quantify how far apart objects are, and temporal measurements are used to quantitatively compare the interval between (or duration of) events. Although space and time are held to be transcendentally ideal in this sense, they are also empirically real—that is, not mere illusions.
So clearly Kant is a realist regarding space and time similar to Newton's absolutism of space (he defended Newton in his works), not an idealist such as Leibniz's space relationalism. This is consistent with Kant's famous synthetic a priori position regarding space and time, which are verifiable independent of anyone's experience under this POV. Currently most versions of mainstream physics spacetime notions belong to realistic POV.
Let's imagine everyone viewed the world, for the sake of argument, through hyperbolic geometry rather than Euclidean geometry. Then our mathematical and physical models of the world would have to begin with that fact because that is the ground of our observations, no matter if later we found that in some way Euclidean geometry was a better fit. It's in this way that our physical and mathematical models are mind dependent.
What you are trying to do with such question is to oppose physics to perception. That is, assuming that space and time would be, from a Kantian perspective, transcendentally ideal, which existence depends absolutely on the mind, but from a scientific/physical perspective, a reality independent of the mind. In such context, you are asking for the relationship between both.
To start, science seeks for empirical truths, not final truths. When we make science, we know that the product (scientific knowledge) will be dependent on our perception, that what we come to learn is not a final truth. In such sense, any scientific approach (or any formal approach from physics) of space and time is assumed to be dependent on the mind. Finally, the physical perspective of time and space has the same basis as the Kantian perspective. So, they are (necessarily) logically coherent:
The scientific and the Kantian views of space and time are essentially the same.
In order to find the relationships between the scientific and intuitive perspectives of space and time, I will use an analogy, comparing the literary and emotional perspectives of a feeling. Both can be equivalent in the sense that they would just be two different forms of an intuition.
So, what are the relationships between the scientific and the Kantian perspectives of space and time? They are the same as the relationships between a feeling and an artistic work expressing such feeling (e.g. the relationships between joy and a poem evoking joy). The description of time and space in physics would be just a formal way to express perceptions of space and time. The only difference is that formal physical descriptions use mathematics to describe experience, and subjective intuitions of space and time are just the product of our subjective experience. But mathematics provides an additional advantage: math is not only a language to communicate concepts, like english, but also a tool that allows analyzing such concepts. So, math provides intuitions of new forms (relativity is essentially a different expression of our intuitions of space and time), like poems can provide feelings of a literary form, which would be essentially different expressions of the same intuition.
So, relativity would be essentially a set of formal conclusions raising from intuitions and empirical concepts, that are the product of a mathematical analysis. Relativity would not be possible without mathematics.
Perhaps you are probably still curious about the effect of the formal, scientific knowledge, over our intuitions. For example, questioning what does the relativistic spacetime curvature implies over my intuition of space and time? In the analogy, how does a poem of joy impact on my intuition of joy?
The answer would be: in nothing. Following the example, there would not be any qualitative change on the effect that a poem of joy evokes on my feelings of joy. In order to know any possible effect, noumenal knowledge would be necessary, so to compare it with phenomenal knowledge, to observe the relationships. Following the example, if joy would be the equivalent of a white light from a noumenal perspective, we could say that the relationship that poems have with feelings is that poems of X specific class have the capability of evoking white light on a phenomenal perspective.
Kant's notion of space is a transcendental one. Which means that space has an autonomous existence but it transcends our perception of space, or time, for that matter, as irreversible perceived changes of objects in perceived space. We will never, according to Kant, be able to know the transcendent reality an Sich. Which is a kind of Platonic notion, because Plato thinks the same about mathematical objects, existing in an extramundane mathematical heaven. We can perceive math, like spacetime, while not knowing how the math objects themselves look like.
In his left glove thought experiment, Kant argued that space is not relational as Leibniz stated. The relationships between the parts of a left-handed and right-handed glove are the same and still they are different gloves. So space is not relational. Which doesn't mean that space can be perceived without looking at objects that stand in a relation to other objects. Space and objects in it have a mutual dependence though. Space without objects is a, well, empty notion.
So Kant believed (wrongly, in my opinion) that space refers to something autonomous, independent of creatures, but it isn't space (or time) as we see it. As the porridge in which objects are submerged as dried raisins.
He would have regarded the modern notion of spacetime as a perception referring to an autonomous thing, substance, essence, entity, or whatever you wanna call it. But the perception isn't the real thing, which we will never be able to contemplate as we're bound to perception. Which of course can be questioned. We might perceive a transcendent reality.