You're exactly right, its a weak acid that won't quickly cleave TBS, but allows you to protonate your bases and get them into the water layer. HCl can hydrolyze your silyl ether, though TBS is pretty tough and if its low concentration HCl for a short time you shouldnt lose much yield. You could always make your own from ammonia solution and HCl.
That said, I've always had excellent results 1:1.1:2.2 eq of alcohol:TBS-Cl:imidazole in DCM, THF or DMF at RT, overnight under N2. EJ Corey in the 70's found that imidazole is uniquely excellent at TBS protection. The one catch at those ratios is that sometimes you get a little TBS dimer and have to distill or run a hexane column. In your case though, you can just do 1:0.95:2 ratio, and leftover phenol will be water soluble. Should get very high yield, and the workup is just water since imidazole is quite soluble in water.
I'll go ahead an toot my own horn with the procedure in my supporting info http://onlinelibrary.wiley.com/doi/10.1002/anie.201506253/abstract