Hi Pieter,
I believe Your trouble comes from capacitor modification at the output of LM7805.
Transients on power supply rail are occurring during TX/RX switching.
As You wrote - You have swapped electrolytic capacitor with 4500uF - as I understand at the output of LM7805.
This cap is too large for this linear regulator and is not reducing heat of LM7805. At higher frequencies this cap has already high impedance and not filters peaks/transients enough - it's passing them into 5V rail (causing arduino reset). Please add additional 100nF capacitor at output of LM7805 or better replace 4500uF with 10uF and obligatory add mandatory 100nF capacitor or return to single original (was it 1uF?).
?
Regards
Tomasz
?
?
tomekdm sp7etd