Conditional sentences - which is correct?
Solution 1:
First, allow me to make a little tweak to your two versions, by using the past-participle "gotten", so that my AmE ear can help me out:
If I had bought insurance for the trip, I would have gotten a refund after I got sick and had to cancel.
If I had bought insurance for the trip, I would have gotten a refund after I had gotten sick and (had) had to cancel.
Yes, your evaluation "I feel the 1st one sounds more natural" is good.
Both versions are grammatical. But version #1 uses the simple preterite where version #2 uses the compound preterite-perfect -- and so, version #1 would often be preferred because of its greater simplicity. (2002 CGEL, page 158, within section "(e) Simplification".)
LONG VERSION: Your question is interesting, for it involves many issues (besides the issue of BrE vs AmE):
remote conditional (which some traditional grammars think involves subjunctive modality)
simple past-tense versus past-perfect construction (i.e. preterite vs preterite-perfect)
optional backshifting
The context for your situation is that you had not previously bought insurance, and then you had gotten sick and had to cancel the trip. Thus, your conditional is a remote conditional construction (not an open conditional), and since the conditional "if P" part is counterfactual, many traditional grammars would consider that subjunctive modality is involved.
Your question is about a subordinate clause that is within the "then Q" part of the conditional construction: that is, does the subordinate clause "after I [got/had gotten] . . . cancel" have to be backshifted?
The answer is no. In your example, the backshifting is optional: version #2 does that backshift, but version #1 doesn't. When the backshift would replace a preterite (that is, a past-tense verb) with a preterite-perfect construction (that is, a past-perfect), then often the speaker has the choice which one to use (but not always).
Because the version with the preterite is simpler, then there's usually a preference for that simpler preterite version (instead of the preterite-perfect). And so, for your example, there is a preference for version #1.
Caveat: This explanation is specifically tailored to your specific examples; for other types of examples, the evaluation and explanation can be significantly different.
For more info related to this subject, there's the vetted grammar source: the 2002 reference grammar by Huddleston and Pullum et al., The Cambridge Grammar of the English Language, pages 155-158 (especially the discussion about [32] on page 158).
Solution 2:
I'd say:
If I had bought insurance for the trip I had to cancel due to illness, I would have gotten/received a refund.