According to Proposition 3.9 of Ozsvath and Szabo's original paper there are indeed topological conditions one can put on the homotopy class of discs to ensure that one can take a single almost complex structure. Moreover, there's an explicit complex structure (coming from a symmetric product of a complex structure on the Heegaard surface) for which you can achieve transversality.

**Edit:** On a second (more careful) reading of your question I realise you already knew this. Apologies. More pertinently to your question, if you have a regular almost complex structure then (provided you're only interested in finitely many homotopy classes at a time) you should be able to perturb $J$ slightly and it will remain regular (for each homotopy class regularity is an open condition). Certainly as long as it's tame you'll never run into problems with Gromov compactness.

**Disclaimer:** I know very little about Heegaard-Floer so maybe someone more specialised can say something more helpful and direct. Instead let me say something more general about transversality for holomorphic discs.

In general in Floer theory you need domain-dependent almost complex structures to achieve transversality for holomorphic discs/spheres.

The problem is that when proving transversality you make perturbations to the almost complex structure and if the disc is multiply-covered in some region then you may end up having to make different perturbations at the same point in the ambient manifold (to which different points of the disc are mapped). Open Riemann surfaces are particularly bad in this way because different regions can have different covering multiplicities (just think of something like Milnor's doodle).

By contrast, a closed holomorphic curve has an underlying simple curve for which one can prove transversality. Of course, for closed spheres in Calabi-Yau 3-folds, the branched covers of a simple, regular sphere are themselves not transverse (they necessarily occur in high dimensional families by varying the branch-points) and you need domain-dependent almost complex structures (or abstract perturbations) to cope with this (e.g. to prove the Aspinwall-Morrison formula)

For discs there is also a theorem of Lazzarini (probably in one of these papers) which lets you decompose a disc into subdiscs which are multiple covers of simple discs and this is how people using monotone Floer theory usually cope with the problem. Of course if you have Maslov 0 discs you run into the same problems as for the Aspinwall-Morrizon formula, hence the monotonicity requirement.