I don't think it is possible for the situation when you are writing a new e-mail and pick the address from the "Contacts".
A workaround: put that "extra" information in the "organization" field in your contacts. That field is searchable the same way as the name field (it works on my Motorola phone), but Aquamail will not include that field.
You can also use the field "Title" for the same purpose.
Yet another workaround that seems to work on my phone. If you start from "Contacts", by finding the contact, and touching the e-mail address, and choosing Aquamail to send a message, it doesn't put the name, only the address. (But I didn't test this carefully by actually sending the message.)
For the situation when you are replying to an e-mail, and want to avoid insertion of the information about your contacts, you can disable that in Settings -> Contacts -> "Replace Contact names".
Note that there two options like that on that screen, and one has to guess what "except on the compose screen" means (for the first occurence). I believe, that's only the substitution for when you are in the messages list or reading the message. But the only way I know that is because I saw that request in one of the threads, and Kostya responded recently that it was implemented.
@Kostya, It is the 1st (or maybe the 2nd) time I am looking at these two looking almost identically settings, and it is cryptic.
I understand that you may have chosen the wording to make it easy for translation...
For the 1st occurence, I'd suggest to make a positive description, not the negative one. (I.e. not where it doesn't control, but where it does)
How about "Replace original contact names with those from your Contacts for viewing messages only"?
And maybe even the option itself: "Replace contact names for viewing".
And the second one, - "Replace contact names in replies"
The description of the second message is also rather confusing: is it replacing only "on the compose screen" (as it claims), or also in the actual messages? I assume, it is the latter. Hence, I'd suggest changing the description text:
"Replace original contact names with those from your Contacts in reply" (or "... when replying")