£100 difference between Y and PE for even just the DOH-LHR leg?? I think you'd find most people here biting BA's arm off for that cheap an upgrade!
I'm also unsure why economy would be giving you free seat selection if premium economy doesn't. Unless you were getting results for full fare Y economy, which would be pretty darn expensive!
However it does sound like economy might simply be sold out on the DOH-LHR flight, but someone with access to the flight stats will be able to confirm.