Here in the UK, we don't expect Chinese-speakers should have an English name (although of course we know that many do). Therefore, if you said "my name is Tweety" but we knew you had a Chinese name, then I think we'd usually treat Tweety as a nickname and Chia Yin as your "real name". "Tweety" might raise eyebrows or attract a few comments, but ultimately nobody cares what your name is and they're more likely to be pleasantly surprised than to be confused or think of it as unprofessional. Then again, most people here in the UK don't choose their own name, they stick with what their parents gave them or a common variant ("Steve" being short for "Stephen" in my case), so we try not to judge people by their names too much.
So from an English-speaking perspective, if you were in the UK, I'd say there's no problem at all with putting "Chia Yin" on business cards and job applications, and then asking people to call you Tweety once you're on first-name terms.
Of course, you're not in the UK, and the tradition of having an English name is much stronger in other places. If choosing your English name is considered part of how you present yourself in a business context, then I'd agree with others that in English Tweety will be associated almost entirely with the cartoon character, so that's how you're choosing to present yourself.
So yes, it's unusual and it might well seem ridiculous to some people at first. If you need a "normal-sounding" English name for business purposes, then Tweety isn't it -- that's a matter of (lack of) common use. It's a matter of business opinion whether your English name needs to be "normal-sounding" or not, but if you're in Taiwan and your teacher is Taiwanese, then presumably there's something behind that advice. You should probably consider what the conventions are for Taiwanese business more than what they are in English-speaking countries, because the most important people judging your English name might well not be native English speakers anyway.
There's no reason you can't continue to use Tweety as a nickname and also have an English name and your Chinese name. Then you're free to decide whether to introduce yourself to colleagues using your nickname or not, once you have your own business experience to draw on. If you choose not to, there's a lot to be said for having a separate "business mode" and "casual mode". I've known English people who use different names at work from with their friends. It's not the norm, and it might make people think a little harder when those worlds collide, but it's nothing people can't deal with.