SHORT ANSWER
Go talk to China Telecom/Mobile/Unicom about Data Center hosting of your dedicated cloud storage - regrets - I don't have direct contact info, but ask around. This is your best long-term solution - although it incurs a recurring cost in addition to training and initial setup costs. They might (highly doubtful) even know more about this stuff than me...
LONG VERBOSE ANSWERS FILLED WITH TECHNO-JARGON AND USELESS TRIVIA
T1 is predominantly copper at roughly 1.5 mbps dedicated. T1 Fiber is archaic and hasn't been around for decades. Most fiber runs at 100mbps to 1+gbps (1,000 x 1mbps) speeds - example - FTTH (fiber to the home). Current ADSL can achieve 8-10mbps in a shared environment (up to 250+ commercial/residential users - all watching IPTV/PPTV/youku, etc). My wife and kid BOTH watch their own respective videos on their own respective computers at the same time - making it difficult for me to shop on taboo and write idiotic verbose, nonsensical responses on gokm.
Bopping out to HK/TWN may require a dedicated line (ultra expensive - which obliterates your < USD 10k capital requirement. I'm assuming USD 10k is a capital (one time) expense, NOT a recurring (annual) expense. If not a dedicated line - you will DEFINITELY need to spring for a VPN - which seriously ramps up and down on speed depending on time of day (internet and IPTV/PPTV/youku/tudou etc traffic patterns) - however it allows you to host off-shore - but makes access from Kunming frequently hellish and frustrating, not to mention intermittent, temporal, inconsistent connectivity issues thanks to firewalls, blackholes (network term), and solar flares (yes, solar flares affect the internet).
You specified multiple TB - so you're in the realm of RAID systems (high availability, failure resistant, allegedly inexpensive - but that's a matter of perspective). A raw 4TB drive runs about CNY 1k these days - RAID 5 requires 3-5 of these drives (5 being better than 3 for various technical and operational reasons). Cloud storage basically means a certain amount of dedicated storage from a RAID farm, with dedicated or share IP addresses/domain names pointing at that storage carve-out....much easier just to say cloud...but it's a marketing word used to describe old tech, used in a novel, innovative, and cost-effective new way...so I go with it...
Your best bet is to go talk to a data center operator - China Telecom, Unicom, China Mobile, and ISPs, such as Great Wall, AIPU, etc. as noted above in SHORT ANSWER. This is also the optimal solution if no-one in your team knows doodle about network storage, cloud storage (same thing), routing, remote access, AAAs (authentication, authorization, and accounting), etc etc etc on high tech internet stuff (technical catch-all word). I haven't touched this stuff for over 10 years, so I'm really rusty and you can't afford to pay me anyway - my consulting fees will similarly obliterate your budget.
If your team isn't highly mobile - a local data center operator is best - traffic just moves so much faster when you operate within the borders of China...as per SHORT ANSWER.
Opting for online storage (along with a domain name, etc) comes with recurring costs - but they (allegedly) will also take care of maintenance and operations (daily, weekly, monthly backups) and network security (admin & operations). Anyone in China breaches your system - they'll have to deal with the government - and the government in China is REALLY REALLY REALLY good at nailing domestic cyber criminals.
If your team/clients need international access - then it depends on who are the traffic hogs/primary/frequent/VIP users and you should POP (Point of Presence) accordingly.