I am posting an answer to my own question, so that it will be marked as answered and may possibly resolve a similar confusion for someone else, in the future. This question was of the form "People keep saying that something goes wrong and I can't figure out what!" So the answer of "Nothing goes wrong" from Tom Goodwillie was necessarily quite brief, but still very much appreciated.

**First, the main point:** $K$-theory is a well-defined cohomology theory on the category of finite-dimensional CW complex, which can be described by virtual vector bundles in the same way as it is for compact Hausdorff spaces (again, I am just doing this for complex $K$-theory, but it should be similar in other cases). In the compact Hausdorff setting, we use compactness to conclude that any map $X\rightarrow BU$ factors through some $G_n(\mathbb C^k)\subset BU.$ In the setting of finite-dimensional CW complexes, we instead use cellular approximation (since $G_n(\mathbb C^{2n})$ contains the $n$-skeleton of $BU$). The fact that we can always homotope to a map $X\rightarrow G_n(\mathbb C^{2n})$ (for a suitable $n$) also means that any bundle over $X$ is stably equivalent to one of dimension $\leq n$. This all can be phrased in terms of the "bundle dimension" discussed in Exercise 3.3 of Husemöller's *Fiber Bundles.*

The natural isomorphism of functors $\tilde K^0(–)=[–,BU]$ (where we can also talk about ring structure) is the only needed connection between classifying maps and virtual bundles. After that, any $\tilde K^i(X)=[X,\Omega^{|i|}BU]=[\Sigma^{|i|}X,BU]$ is described in the same way in terms of virtual bundles over $\Sigma^{|i|}X$, which is still a finite-dimensional CW complex.

**So what ***didn't* go wrong? As I mentioned above, I could only find one explicit mention of things possibly going wrong in finite dimensions, which was given by Dmitri Pavlov here. While my understanding is still limited (e.g. I don't know what a Grothendieck topology is), I think I understand the basic idea of his answer (which I'm quite grateful for, as I've never explicitly thought about $K$-theory from a sheafy perspective before).

The idea seems to be that virtual bundles form a presheaf and the question becomes "Is this a sheaf?" As usual, it seems that locality is straightforward, while gluability requires some more thought. We can piece together two virtual bundles that are compatible on their overlap, but given an infinite collection of open sets, the dimensions of the bundles may be unbounded. This is a big problem for gluing. But if we have a compact base, then we can restrict to a finite subcover, after which gluing goes through fine. If our base is a finite-dimensional CW complex, then we may not have a finite subcover, but our virtual bundles are stably equivalent to bundles of bounded dimension. Thus gluing still works without any issue of unbounded dimension.

**Lastly, what ***can* go wrong? In this answer, I don't want to give the impression that $K$-theory doesn't face issues in being generalized outside of the compact setting. Certainly, it does! My point is that, over CW complexes, the issue is infinite-dimensionality rather than non-compactness. Over infinite-dimensional CW complexes, the functorial correspondence $\tilde K^0(–)=[–,BU]$ (where the left-hand side is defined in terms of virtual bundles) breaks down, because maps into $BU$ no longer need to factor through a finite stage. Perhaps the simplest example is the inclusion $\mathbb C\mathbb P^\infty\subset BU$. We can still try to define $K$-theory in terms of virtual vector bundles, but it is no longer described by this spectrum. In fact, over infinite-dimensional CW complexes, this $K$-functor does not satisfy the Eilenberg-Steeenrod axioms, so it is *not a cohomology theory* and is not represented by any spectrum.

For more information on issues with infinite-dimensionality than I am able to provide, you can check out "The $K$-theory of Eilenberg-Maclane spaces" (Anderson and Hodgkin), which I got from Mike Doherty's answered to the afore-linked question. Alternatively, in the comments to this question, there is a link provided by user43326 to a postscript file of "Vector Bundles over Classifying Spaces" by Bob Oliver. A longer paper on the same topic, by Oliver and Jackowski, can be found here.