Thanks Jason.
I did what you suggested. The 150 first emails are sent, then 500 error again.
The SMTP server is indeed shared. That sure impacts performance in delivery speed, but not the daily SMTP relays limit for that particular domain, which is 250. 200 is what I decided to keep a few relays for other forms (no one else is using the site at the moment, so I didn't reach that limit through the other forms...)
The error does not occur while sending, but when it should be «paused».
On every tests I did, whatever Burst size & time and Wait values I specified, the number of emails I specified for the first batch (may it be 1, 10 or 200) was always sent successfully. Depending on the values I entered, the process ran successfully for periods varying between 1-2 minutes to over 40 minutes. It is 4-5 minutes after entering what should be «paused status» (while not sending anything to respect limits) that the browser pop the 500 error page.
I'm pretty sure the 500 error comes more from the idling php page rather than reaching the SMTP limit. But that said, even after turning on all php error loggings possible and adding the 3-4 php lines you suggested in other threads, GoDaddy's Apache and Error logs are still completely useless.
What is the way to process the php file containing the UE behaviour, start the sending process, and redirect user to a confirm page ?
