Probably not: it's written for melee weapons (the throw should work)
Here is the text of Sword Coast Adventurer's Guide it indicates that this style of combat is used with swords.
Under the blades
They are elf magicians who master a sword fighting school based on a tradition of arcane magic. (SCAG, 141)
Under the green box (Blade Styles) on the next page:
Since its creation, as a martial and magic art, Bladesinging has
attached to the sword, specifically the long sword. (SCAG, 142)
Other melee weapons can be used with the blade singing.
When you read further in Bladesinging
Cat: The styles that use a sword belong to this family …
Bird: Styles that focus on the use of a handmade weapon, such as an ax or a hammer, have been grouped together as bird styles … Eagle style blades use small hand axes and many maneuvers in style focus on fluid manners throw the weapon and draws a new one.
Snake. Practitioners of these styles use a flail, chain or whip … (SCAG, 142)
Nowhere does it mention crossbows, or any other bow. If they had intended to include one-handed bows in this style, they would probably have been placed in this green box with an animal name. Given the throwing hammer style, a stab style launched would be appropriate, perhaps as a variation of the Cat fighting style.
Why does not the Level 2 feature say "melee weapon" like the level 14 feature does?
Good question. I will tell you that after writing all of this on the swords in the introduction of the Bladesinger class and on the named animal in the green category of some other melee weapons, the authors did not say anything about it. have not felt the need to be redundant in their phrasing. of the 2nd level class which opens the Bladesinger School. Or they forgot about it during the last edition and now regret the day when the minors of text discovered their forgetfulness. 😉 The evidence shows that this is related to melee weapons and not to any type of bow.
Ask your DM
If your DM is ready to let you use a handbow with the blade singing, you will have to work together and manage the level 14 class function when you get there. I hope your adventures will plunge you into the countryside!