Let me argue that Kunen's argument actually shows the best possible thing here.
First, let's think about consistency results. The "ideal" result here would be:
(i) Con(ZFC) implies Con(ZFC + "There is a stationary-reflecting, non-weakly-compact cardinal.")
Obviously we can't prove this in any reasonable theory (I assume that we try to prove consistency results in something vastly weaker than ZFC) unless ZFC is inconsistent, so the actual ideal result is:
(ii) Con(ZFC + "There is a stationary-reflecting cardinal") implies Con(ZFC + "There is a stationary-reflecting, non-weakly-compact cardinal").
Note that this is a strictly stronger statement than "Con(ZFC + "There is a weakly compact cardinal") implies Con(ZFC + "There is a stationary-reflecting, non-weakly-compact cardinal").
Although Kunen seems to begin with the assumption of the consistency of weak compactness, he's actually proving the ideal result (ii) above! This is because - assuming the consistency of stationary-reflecting cardinals -
there are three cases:
Stationary reflecting cardinals are inconsistent. Done, vacuously.
Weakly compact cardinals are consistent. Kunen then builds a model with a stationary-reflecting, non-weakly-compact cardinal.
Weakly compact cardinals are inconsistent, but stationary-reflecting cardinals are consistent. In this case we're also vacuously done.
The only interesting case is when everything involved is consistent, and that's what Kunen treats. So the apparent use of an additional consistency hypothesis is in fact unnecessary, since it only ignores cases (i) and (iii) which are each vacuous.
Now what about actual models?
Obviously if T is stronger than ZFC, then "T doesn't imply "every B is C"" is stronger than "ZFC doesn't imply "every B is C."" So we can ask, e.g.,
- Does ZFC + [large cardinal] imply that every stationary-reflecting cardinal is weakly compact?
The answer to this is basically no, since basically every large cardinal notion is preserved by small forcing. Start with a model with your large cardinal property holding above a weakly compact - say, $V\models$ "$\kappa$ is weakly compact and there is a supercompacts $>\kappa$." Now apply Kunen's construction; this is a small forcing relative to the supercompact, and so we get a model of "There is a supercompact and there is a stationary-reflecting non-weakly-compact cardinal."
This result essentially says that reasonable large cardinals tend to have no bearing on the implications between various combinatorial properties, since these implications tend to either hold or be breakable by small forcings. So to me this suggests that this isn't really the right place to look.
This ignores, of course, those large cardinal hypotheses which do affect combinatorics, like non-measurable weakly measurable cardinals, but these are a bit odd; similarly, I would argue that generic large cardinal hypotheses a la Foreman aren't really large cardinal hypotheses.