Can I use Mentor of the Meek on small guys with buffed stats?
No.
Both abilities of Vanquisher&#39;s Banner and The Immortal Sun that gives creatures you control +1/+1 are static abilities.

611.3. A continuous effect may be generated by the static ability of an object.
611.3a A continuous effect generated by a static ability isn’t “locked in”; it applies at any given moment to whatever its text indicates.
611.3b The effect applies at all times that the permanent generating it is on the battlefield or the object generating it is in the
appropriate zone.
Example: A permanent with the static ability “All
white creatures get +1/+1” generates an effect that continuously gives
+1/+1 to each white creature on the battlefield. If a creature becomes white, it gets this bonus; a creature that stops being white loses it.

But what comes first when the 1/1 creature enters the battlefield zone. The trigger from Mentor of the Meek or the static abilities?

611.3c Continuous effects that modify characteristics of permanents do so simultaneously with the permanent entering the battlefield. They
don’t wait until the permanent is on the battlefield and then change
it. Because such effects apply as the permanent enters the
battlefield, they are applied before determining whether the permanent
will cause an ability to trigger when it enters the battlefield.
Example: A permanent with the static ability “All white creatures get
+1/+1” is on the battlefield. A creature spell that would normally create a 1/1 white creature instead creates a 2/2 white creature. The
creature doesn’t enter the battlefield as 1/1 and then change to 2/2.

The 1/1 creature enters the battlefield as a 3/3, given that it's the type needed for Vanquisher&#39;s Banner. So Mentor of the Meek won't trigger.