Don't give up on that flascart just yet, you still have a couple of options:
1.If you have a DS/DS Lite or can borrow it from someone, try your flashcart with it - if it works with DS, getting it to work on a 3DS should just be a matter of finding the right launcher files.
2.Flash the latest firmware of your flashcart (I'm not talking about the Wood R4 launcher files that you put on your microSD, but an update file that you launch with a DS). Don't be afraid to try the launcher files from different websites, despite that it says
www.r4isdhc.com.cn on your flashcart, you can try other websites like
http://www.r4isdhc.com/ or
http://www.r4isdhc.hk/ or
http://www.r4i-sdhc.com/ I'm using the Wood R4 files downloaded from a different site than printed on my flashcart.
3.Try different launchers posted in
Apache Thunder's TWL flashcart launcher thread. If not Wood R4, perhaps a Moonshell 2 or the DSTT launcher will work for you. I know for a fact that DSTT launcher works with my flascart just as well as Wood R4, I'm just using Wood R4 because I like its interface better.
4.If you are running a Custom Firmware on your 3DS (you are able to install .cia files), you might want to try
Apache Thunder's TWL flashcart launcher. With that launcher I'm able to use the very old, original R4 with my 3DS (and that thing never even supported the DSi).