How can I select all files of a type in a non-Finder, application window?
Here is the scripting definition file for the latest version of Transmit, downloaded today from the Panic homepage.


  Transmit.app AppleScript Terminology


Conclusion: Having read through the terminology and running a few brief tests in Script Editor to pass the time in an Apple Store, I highly suspect it is not possible to use AppleScript to select files in a Transmit file browser.

Justification: Most notably, there is no command that is called select, or with a name indicative of a similar function. Also lacking is any property belong to any element or class object that sounds like selected or selection.

There is a class of object belonging to a file browser called a selected browser item. This, as you may be able to guess, is an object reference to the currently selected file(s) in the file browser. Therefore, it is possible to retrieve a list of files currently selected:

tell application id "com.panic.Transmit"
    tell the front document
        if not (exists) then return null
        tell the current tab
            tell the local browser
                get the name of every selected browser item
            end tell
        end tell
    end tell
end tell


Any attempts to set the selected browser items either collectively or individually by index was unsuccessful. 
 Invoking the make command to create a new selected browser item also failed.

Sorry that this might not be the answer you were hoping for.