I can't see any way you're going to be able to do it - even ignoring the rules...

The problem is that T1 is further away from the city than T3, so it's not like you're just stopping in at another station on the way - you're actually heading out of the city (for one stop) and then back into the city.

In my experience the airport stations are fairly strict with checking tickets. Many other stations will let you avoid the gates if you're carrying luggage, but the airport stations have wide gates to allow for luggage so they make you use them.