There is a method using the single screen warp glitch on the non dx version but it does involve sequence breaking. You can use the glitch to get the level 3 dungeon key. Also use the glitch to enter the castle and get the 5 golden leaves which causes the game to think the monkeys built the bridge which in turn causes tarin to appear by the beehive. talk to him and he asks for the stick you're carrying which is actually the bananas lol. from there continue the game as normal.
Go to the very edge of a screen that Link can walk to another screen from. Move towards the edge of the screen and just before it begins to scroll press "Select". Press "Select" again to get rid of the map and Link should now be on the new screen but in the place he was on the old one.
NOTE: This glitch has been fixed in the DX version and in some later revisions of the GB cart.
So long as Link does not speak again to Madam MeowMeow, he will be able to keep Bow-Wow for the rest of the game. Also, if the game is otherwise beaten normally, Link will magically have the stick when Tarin asks for it, even though Link never actually obtained it. Asking Marin to follow you to the Walrus while having Bow-Wow causes her to become a glitchy version of him. If you enter a building and exit, the real Bow-Wow will be gone. Talking to Madam MeowMeow turns Marin back to normal and returns Bow-Wow to his post.
If it can't be done on the GBC DX version, why post it on the GBC DX version board? It doesn't belong here.
Take a look at the GB version board for your answer.
TC could have been clearer for those who are unfamiliar with the glitch, but I'm sure you can see that there are one or two similarities between the GB and the GBC DX versions that would make people who are interested in one game also interested in the other...
Aren't there some characters who refuse to talk to Link if he has Bow Wow? I'm pretty sure Richard is one. (although if you're using the Screen Warp trick anyway, you can get the key without talking to him)