I use ThinkorSwim. The Level 2 / T&S isn't the best as it lags a little sometimes. 

I figured that to overcome this potential issue when it comes to getting the best execution, I could pay for some Market Data Service to get a better quality Level2 and tape that doesn't come with TOS. That way I can see the correct data in the moment.

Unfortunately I don't have a huge account, so I can't change brokers yet. Especially to one that offers the best in terms of data and execution. 

Now the question is this: Let's say I see the Ask at a price that I am willing to pay for on the non-TOS (more accurate) Level 2 data. The TOS level 2 shows a different price on the ask due to lag, but I send a Limit order for the best ask right at the moment I see the better price on the non-TOS Level2.

Would I be more likely to get executed at the price displayed on the more accurate Level2? Would I have "beat" the TOS level 2?

Hopefully I am making sense. I am trying to see if paying for more accurate data would allow me to send the order at the right time to most likely get filled at that price I want even though the data the broker's depth provides is lagging by a few seconds.  

thanks in advance!