It looks valid to me. Rewriting it a bit for clarity
with regards to SJO-xMIA-LHR, rule 4(f) states:
4(f) Only one international departure and one international arrival from/to the country of origin permitted
EXCEPTION: Two permitted for origin USA when one arrival-departure is a transfer without stopover.
NOTE - travel between USA and Canada is not counted as international.
So your itinerary complies with the exception - origin USA and your second arrival (from SJO) is a transfer without stopover
And just by the way your first visit to MIA can be stopover if you want
with regards to LHR-xHKG-AKL-xHJKG-ORD, rule 4(e) states:
4(e) Only one intercontinental departure and one intercontinental arrival permitted in each continent except as
follows:
1. Two permitted in North America when one is a transfer without stopover.
2. Two permitted in Asia when one is a transfer without stopover or on direct single plane service between the
Southwest Pacific and Europe/Middle East.
...
Since both of your visits to the Asian continent are transfers without stopovers, you are certainly complying with the second dot point in this rule
And again just by the way, one of your visits to HKG could be a stopover if you want.
The online tool is very buggy. i suggest trying:
- Entering SJO-LHR instead of SJO-MIA-LHR, sometimes the tool manages to handle correctly connections it itself works out
- Try AKL-LAX-DCA instead of AKL-HKG-ORD
Finally, did you know BA flies SJO-LGW (London Gatwick) a few days per weeks?