Larger syringe would be better.
The trick is to "prime" ie fill the slapshot by putting liquid in from an already filled syringe rather than trying to suck it into the empty tube with an empty syringe, then you do not get air bubbles, and you are right, you need a syringe that will hold the dose + the tubes worth. You ought to find out how much your slapshot tube holds by using water, then you will know for future.
Then with a larger syringe simply connect a needle to the syringe and draw up the amount needed + the amount the tube holds (eg if the tube holds 2 1/4 mls and you need to dose the pig with 3mls, then draw up 5 1/4 mls). Then disconnect the needle and put the slapshot into the middle, and full the tube with the excess - bingo syringe with 3 mls ready to inject.
In the meantime if you only have a 2ml syringe and a friend (easier to do with 2 people)
1. Put needle on syringe, and draw up 2 mls into syringe (the max it will hold)
2. Leave needle in bottle (now called the clean needle) and unscrew the syringe
3. Put slapshot (without any needle) onto syringe and inject the two mls into the tube (tube takes about 2/12mls)
4. Put finger on the end of slapshot and remove from the syringe (don't put finger on end and liquid will run out)
5. Reconnect syringe to clean needle and draw up a further 2mls
6. Disconnect from clean needle and reconnect to slapshot (which you have been holding with your finger on)
7. fill remainder of slapshot tube (abotu another 1/2 ml)
8. Note how much is still in syringe
9. Put on second needle to slapshot, and inject pig
10. Remove dirty needle and connect slapshot to clean needle and draw up the extra into syringe
11. inject remainder into pig