There are definitely some strange things happening over here too. First of all, I've been trying to reproduce your errors since yesterday, but I couldn't. Link was at no time invisible and bypassing the menu only works if I quickly turn my GBC back on after powering it off. It won't work on my GBAs.
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software: v3.01
rom: Legend of Zelda, The - Link's Awakening DX (U) (V1.2) [C][!]
hardware: GBA SP (E), GBA SP (J), GBC (E)
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After succesfully flashing the cart, I named my character 'DDD', then saved and turned off the GBC after playing a bit. When turning it back on the save was still there and working, but as soon as I took the cart out of the GBC and put it back in, the character was named 'DDA'. This was happening every time I created a save named 'DDD' when taking out the cart for the first time (doing it more than once had no effect, though).
After I put the cart in my GBA SP (E), the save was gone (this happened only once). The other times I tried the save was working fine, no matter what gameboy I used, where I saved or which slot I chose.
Today I turned on my GBC to see if the save was still there, but it was gone. I created a new save with the name 'ZELDA' (just because I like the alternative music). After that, I took the cart out of the GBC, waited a minute or so and put it back in. Then, after turning it on again, there were two saves. The one I created and another one in slot 3 with all 14 hearts, an unreadable name and 230 deaths. The save worked and, surprisingly, I started at the beginning of the game (lying in bed) with 88 rupees and 14 hearts already there. When leaving the house, that strange dog, on the screen to the left, was already gone and shortly after picking up my sword at the beach, I realized that I somehow managed to get 999 rupees. After saving for the first time, the hearts were gone, second time, they were there again ...
Long story short, I erased the whole cart using only the E-Merger itself and now it seems to work. Interestingly, the writing time for the first flashing process after doing so, went up from about 42 seconds to 273 seconds.