Problems relaying/rewriting

I'm the sysadmin of a small company that is part of a largish intl company
and the last holdout of Unix amongst a Windows-world. Our mail addresses
are of the form user@bigcompany.com, while out local domain is small.local.
All our outgoing mail (except a for few Outlook users) goes through a
local relay, currently a solaris machine running sendmail. It's working
more or less, but I need to switch to something newer. The designated
successor runs Linux and postfix. I'm a total novice when it comes to MTA
configuration, but postfix looked way easier than sendmail!

To make a long story short, I succeeded halfway by doing:

mydomain = small.local
myorigin = bigcompany.com
mydestination =
relayhost = [mail-relay.bigcompany.com]

sender_canonical_maps = hash:/etc/postfix/sender_canonical
sender_canonical:
@small.local @bigcompany.com

This shall do:
Relay *all* mail to mail-relay.bigcompany.com.
Replace *all* occurences of "small.local" with "bigcompany.com"
(sender/recipient, envelope and header).
Add "bigcompany.com" to recipients without a domain.

I also need an alias for "all employees". This I do with a
virtual_alias_map (which is probably totally wrong). Anyway, this
works.

*But* there are several remaining problems: We have a very heterogenous
mix of machines, ranging from SunOS 4.1.4 to HP/UX 10.20, Solaris 2.5,
2.6, 8, Windows 95 to XP and diverse Linux boxen. While most users use
Netscape/Mozilla and set the From:- and To:-addresses complete with domain
part, system programs on those Unix hosts don't. They send mail to
user@small.local or user@mailhost.small.local and with a From:-address
like host.small.local - which will not be rewritten. There are also 2
hosts from a sister company sister.local whose sender addresses need to be
changed to bigcompany.com. All this I just can't get to work.
As I understand it, the rewriting to "bigcompany.com" should be dome at
their mailhost. Alas, I just can't ask them to do it. If I tried that SMTP
hosts might just got shut down and we'd be advised to "use Exchange, like
all the others". Which means I _have_ to find a local fix. This
environment might be somehow "illegal", but its all I'm going to get.
When I summarize what I need, it looks simple enough, but all those MTAs
refuse to work in a straight-forward, intuitive way. ;-)

Summary:
All recipient domains of the form @*small.local shall become
@bigcompany.com
All sender domains shall become @bigcompany.com
Aliases like users@small.local shall be expanded to user1@bigcompany.com,
user2@bigcompany.com, ...
All mail must be sent to the "smart" host mail-relay.bigcompany.com

I would *really* appreciate any help on this.

Thank you,
-hjw