Note, I use Mutt on an iMac running Leopard but I don't use MobileMe.
I could be wrong, but I don't think your Mutt is sending directly - it
is probably using a local MTA on your machine. Your "mutt -v" output says:
So does mine, but I override that in my .muttrc file with the line:
set sendmail = /opt/local/bin/msmtp
In either case, it is an MTA which is sending the email for you,
presumably in your case sending directly to the recipient's MTA.
I don't think that is a good idea in general because I am lead to
believe that some MTAs are set up to reject email from MTAs on dialup
So what you need to do is set up your MTA (whether that be
/usr/sbin/sendmail or something else) to send via an outgoing SMTP
server that you have access to.
You could presumably use one at MobileMe, but bear in mind that it will
(or at least should) ask your machine to authenticate (to prevent people
using it as an open relay).
If, like me, you only send email from home while connected through your
ISP then you could alternatively use your ISP's own outgoing SMTP server
(since they automatically know that you are one of their customers
authentication is probably not required).
If you are using /usr/sbin/sendmail then you will have to work out how
to configure it.
Personally I prefer to use a "lightweight" MTA, namely msmtp, which I
installed (like Mutt) using MacPorts.