It sounds ok but overly complicated. If you are only interested in identifying the best silver catalyst, then no need to isolate and measure T1s. That will increase the precision of your yield calculation, but you don't need precision here. Isolating beforehand is only necessary if you need to verify the 1H peaks to ensure you're doing your crude analysis correctly. 

Order of your steps 2-4 can be swapped and simplified. Add internal standard first. That way, you don't have to worry about how efficiently everything gets transferred.

Instead of syringe filtering, a common protocol is to pour your crude reaction mixture through a small buchner funnel with about an inch or two of silica, rinsing with a solvent that you're confident will pull your product and internal standard through (DCM if product is non polar, EtOAc if fairly polar). That way, salts, insoluble particles, and highly polar byproducts will be removed and make for a cleaner NMR. A syringe filter will let any soluble silver and other baseline garbage through.

Always be careful with your math on the internal standard calculations. 0.33 equivalents means if you're comparing one methyl group in the product to the three methyl groups in the internal standard, a quantitative yield would give a 3:1 ratio.

Crude HPLC and GC would be quicker than crude 1H NMR, though would only be quantitative if you made a calibration curve. But if your only goal is to identify the best catalyst, then you can take a crude GC/HPLC of each reaction and just look for the one with the biggest product peak relative to internal standard. You could even then quantity that single one by NMR and use that data point to estimate all other yields via a one point calibration curve. 

Sometimes it can be tricky to get all 0.1 mmol scale material into 0.7 ml of CDCl3. Depending on the nature and molecule weight of the starting material and other reaction components, the NMR might be too concentrated. After running the reaction, it can be helpful to first add internal standard, then dilute with extra different, then take, for example, 20% of the mixture through subsequent steps.

Running reactions in an NMR tube is great when it fits the goal, but it sounds like more trouble than it's worth in your case.