You can do this if you're willing to programmatically manipulate hidden configuration variables in your config file ;-)
I use Focus Labs Master Config style configs, so I have this in my dev config:
$env_config['template_debugging'] = 'y';
$env_config['show_profiler'] = 'y';
And, obviously, you can set those to 'n' by-hand to turn off the debugger and profiler.
But, you also could set those values programmatically, inside of your own PHP conditional. In pseudo-code:
$env_config['template_debugging'] = 'y';
$env_config['show_profiler'] = 'y';
if ($currentURL === $myAjaxURL) {
$env_config['template_debugging'] = 'n';
$env_config['show_profiler'] = 'n';
}
I do manipulate site configs programmatically. It's obviously a place where you can accidentally knock your site offline, but if you are sufficiently careful and/or rube-goldberg-ish, the config files are just another place where you can program things via PHP.
If you don't use the Focus Labs Master Config style configs, you still can do this kind of thing in your config.php. You'll just use the proper EE hidden variable syntax:
$config['template_debugging'] = 'n';
$config['show_profiler'] = 'n';
One caveat, though: I don't know if EE will overwrite your code in the config.php if you go into the System Preferences (or, other admin) via the CP. You can always set your config.php permissions to read-only when you're doing this testing, to prevent EE overwriting it, if that does become an issue.