Congrats on your upcoming wedding! I run my own business helping couples with Chinese wedding tea ceremonies - http://www.destinem.com
I've held tea ceremonies at central locations before - like you said, someplace neutral rather than the traditional groom and bride's house can be good if one side isn't Chinese/religious.
Given you've got time constraints - why not do it during the reception dinner? If you're having a 10 course chinese banquet, you will have time during the night to do this. You just might have to cut short some of the other things (eg. long speeches, playing games etc). Unless you want to do the tea ceremony on a different day from the wedding day, I wouldn't suggest finding a 3rd location, because you will need to decorate it, and get guests there etc...this is more logistics to organise.
I've MC'd reception banquets before where I open the banquet by holding the tea ceremony first. Whether this works will depend on how many people you need to present tea to. If it's lots of people (eg. more than 5-6 couples), this can drag on, but if you have someone who is organised and knows what they're doing, then it can be lovely. If you have Western friends who've never seen a tea ceremony before, even better! You can set up the stage with the chairs, flowers, double happiness decorations, floor mats etc and make it really nice. Your MC can do a brief Intro on what the tea ceremony is for (thanking and respecting your parents for their care and love). You will have a tea lady to announce who is getting tea, and she will say nice auspicious words during each tea presentation. So if you have Western friends, it's imperative the tea lady speak fluent Chinese and English for everyone to understand!
If you're having a 3 course western banquet, that actually gives you even more time between courses to do this! If you've booked the venue there should be no reason why the venue refuse your right to do a tea ceremony. It's a little bit showy for your guests, but it can be really special if done well. At the end of the tea ceremony the tea lady announces "you're all a happy family now!", and everyone claps and it's all quite festive! If the bride can change into a chinese qipao, I especially like that part of it.
Anyway, I hope that helps! Happy to give any suggestions if you need more tips!