I would like to use Tortoise Formation to temporarily disable the effect of Ensnaring Bridge. My question therefore is: does that combo allow my creatures to attack, even if there are no cards in my hand?
No, Tortoise Formation has no interaction with Ensnaring Bridge.
Shroud only prevents a creature from being the target of a spell or ability. That is the only thing shroud does. Only spells or abilities that specifically use the word "target" have targets (with a few pseudo-exceptions, chiefly Auras and Equip abilities, for which the word "target" occurs in the official rules but is not usually printed on the card). If a spell or ability does not use the word "target", then shroud has no effect.
Ensnaring Bridge's ability does not use the word "target", so playing Tortoise Formation will not allow your creatures to get around the Bridge's restriction.
You can always use something like Lux Cannon to blow away your Ensnaring Bridge... but you should only do that when you are ready to attack, and know you'll win (or have a backup Ensnaring Bridge in your hand). I'd suggest you use something like Bottled Cloister instead (this will allow you to have a hand during your turn, but remove your hand during your opponents turns.)