Probably Not Strictly RAW Compatible Due to Reloading
First off I must say that I highly recommend the latter half of Nesbitto's answer for explaining why this is a suboptimal choice. I would add only that this might not be relevant to a tier one campaign or to a multiclass dip into Bladesinger (probably not a bad option for an Arcane Trickster or Dex-based Eldrich Knight), and that little stops a character from fighting in one way at low levels and a totally different way when the reach higher levels.
As per the rules you cited, the only things that can end a bladesong are wearing the wrong armor, using a shield, being incapacitated, or "[using] two hands to make an attack with a weapon". While the lore strongly implies having one melee weapon in one hand, there is no such requirement. If you look at this previous question you'll see the argument for them being able to dual-wield, despite this also not really matching the lore. Basically the ability as written only deals with giving limitations not listing valid options, which makes sense in a game where you can do anything.
Lore Based Mechanical Issues?
I hope I'm not being overly contentious, but several answers here seem to rule out the hand crossbow by relying on the fluff and lore of the Bladesinger, citing this question, involving spells not making a rules/fluff distinction, as a basis. This approach seems to be an overly broad reading of the rules on spells (where there is no set apart section with clear rules) into determining that every word of lore relating to character classes is a mandatory aspect of the character and game mechanic. I have never heard of a Warlock character being told she absolutely had to be "driven by an insatiable need for knowledge and power" or that a circle of the land Druid character had to actually spend any of his time meeting "within sacred circles of trees or standing stones to whisper primal secrets in Druidic" but perhaps some tables read such things directly into class mechanics, and presumably they would ban hand crossbow wielding Bladesingers. However I think such lore based limitations are usually viewed more as house rule decisions than a rules as written-based decisions.
For the record, I insisted that my own Bladesinger character had to not only use a sword but literally sing to his sword (the sword was called Caroline, so he sang a certain Neil Diamond hit at the beginning of every combat), but that was obviously very much a self-enforced rule.
But even if all lore is mechanically significant it makes no difference. Even if it is a firm rule that a Bladesinger specializes in and mostly uses a blade, whip, axe, etc., as heavily implied by lore, that in no indicator that they do not pull out a hand crossbow from time to time and that this interrupts their musical magicks.
Thus we are so far in the clear.
Probably an Issue: What does "use two hands to make an attack" Mean?
There is an issue though. As per the equipment entry on bolts: "Drawing the ammunition from a quiver, case, or other container is part of the attack (you need a free hand to load a one-handed weapon)". Thus if loading/reloading being "part of the attack" counts as part of "making an attack" for the purposes of Bladesong then the character has used two hands to make that attack.
While the wording is so close that it seems to strongly imply that this is the case, there is still some wiggle room based on the probable intent of the rule for reloading being to address needing a free hand during the attack action (which can be one of the hands holding a two-handed weapon), and the actual "making" of an attack not necessarily referring to the full attack action.
That said, reasonable minds could certainly differ, and I imagine most DMs would err on the side of handwaving. However under the most conservative reading of rules there does seem to be an issue.
Note: I believe the rules on ammunition were errataed to cover dual hand crossbow reloading confusion, so your Player's Handbook may read differently.