If you use a magically enchanted buckler and a weapon in that hand, do you also lose the enchantment bonus to AC?
Yes, Enhancement bonus is an increase to the item's statistic(s); if you lose the item's functionality for any reason, you also lose any of the bonuses associated.
Essentially a +1 buckler has 2 Shield AC. You would lose your Shield AC by not being able to properly use the buckler, not just the base AC.
For example, having a +1 shortsword in one hand and a regular shortsword in the other doesn't provide you +1 to hit with the second hand weapon. The bonus only applies to the object that is enhanced.
You don’t have an enhancement bonus to AC.
Your buckler has an enhancement bonus to its shield bonus to AC that it grants you.
And yes, this is a confusing mess; see here for more details.
So the base shield bonus is +1, and it gets a +2, say, enhancement bonus, meaning the magical buckler provides a total of +3 shield bonus to your AC.
And since it is all the buckler’s shield bonus, it goes away when you attack with that hand.
The Improved Buckler Defense feat from Complete Warrior can allow you to retain the shield bonus while attacking with that hand, but you still take the attack penalty and at any rate a feat is much too expensive for this. Better to get a +1 animated heavy shield at some point in the future if you want to use both hands for weapons while having a shield.