You found this, presumably: http://talkingmoose.ca/loved-original-talking-moose
Maybe try with Moose 3.1.sit from one of the links on the page above.
The sit file expands to an auto extracting Moose 3.1 Archive, holding all needed files.

As all items from Moose 3.1 Archive are installed, Moose plays as it should in 6.0.8 on my side.

You did install the DA´s - did you place Moose, Moose Phrases, Go Moose and MacinTalk from the Talking Moose 3.1 folder into your System folder as well?
Seems a special version of MainTalk is needed for Moose.

If it helps, I may send you a disk image with Moose 3.1 Archive on it.
Does your SE read HD floppies, btw?

Here is the installation guide from Moose Archive, HTH.

New Moose Installation Instructions

1) Install the basic Moose parts: Drag Moose, Macintalk, and Moose Phrases from the Moose disk to the System Folder of your startup disk. If you use System 7, the Finder will tell you some of the files need to be put in special places. Click OK and everything will be put in the proper places. Drag the “Go Moose” application from the Moose disk to your startup disk. It does not matter where it goes; you may wish to put it with other general utility applications you use. If you only want to do a basic install of the Moose and don’t plan to use the HyperCard stacks, DAs, or FKeys, then you can immediately skip to step 5.

2) Install the Speech DAs: If you are a System 6 user then double click on the Speech DAs suitcase. This will launch the Font/DA Mover so you can install the desk accessories. If you do not know how to use the Font/DA Mover, check your User’s Guide to the Macintosh for instructions. If you are a System 7 user, drag the Speech DAs suitcase file to your System Folder of your startup disk and the Finder will take care of installing the DAs for you.

3) Copy the stacks: If you wish to use the Moose XCMD with HyperCard, copy the Moose goes Hyper stack to your startup disk. The advanced Moose stack contains the documentation for customizing the Moose to speak special phrases in certain applications. Copy it to your startup disk if you plan to become a heavy duty Moose hacker. Put these files with your HyperCard application.

4) Copy the FKeys: If you want to use the Moose FKeys, copy the Moose FKeys suitcase file to your startup disk. The Fkeys in the suitcase file replace previous versions and are updated to work with System 7. You will have to use a utility like Suitcase II, Master Juggler, FKey Mover, or ResEdit to install the FKeys.

5) Restart your computer.

6) The Moose will say hello on startup. Say hello back.

7) Get to the Moose controls: If you are a System 6 user, select Control Panel from the Apple menu. Follow the instructions on page II-2 of the manual. If you are a System 7 user, select Control Panels from the Apple menu. The Control Panels folder will open. Find the Moose icon and double click on it to access the Moose controls. You can now follow the manual from page II-3 on.

Demo Moose and Old Moose users: After you are sure you have made the translation to the new Moose smoothly, you can drag the Old Moose folder to the trash.

Interesting; I just put a Talking Moose video up on YouTube and was chatting with the author last week. You have to have enough memory available to run it. Talking Moose 1.2 came on a tweaked System 4.1 disk where he had stripped out a number of resources in order to free up enough memory on a 1MB system to load the DA correctly.

If you want version 1.2, I'll see if I can get it bundled up in a way that will boot on an SE. 1.2 takes up less resources than 3.1 that we've got on here.