In Spanish and Portuguese, a pesar de means in (de)spite of. But in both languages, pesar has never meant spite – see quotation below on the etymology of spite. pesar hails
From Old Spanish pesar, from Vulgar Latin *pēsāre, from Latin pēnsāre, present active indicative of pēnsō.
How does a pesar de yield the meaning in spite of?
Spite was adapted from Old French despit ‘scorn, ill will’, which was also borrowed intact as despite . This came from Latin dēspectus, the past participle of dēspicere ‘look down on’ (source of English despise ), which was a compound verb formed from the prefix dē- ‘down’ and specere ‘look’ (source of English spectacle, spy, etc). The use of in spite of and despite for ‘notwithstanding’ goes back via an intermediate ‘in defiance of’ to an original ‘in contempt of’.
Word Origins (2005 2e) by John Ayto. p 473 Left column.