I'm starting a Bladesinger character. Is there any way to add the arcane keyword to a weapon so that I may use the White Lotus feats with my basic attacks?
First, an important caveat: there are no weapons that provide power source keywords (martial, divine, arcane, psionic, primal, shadow). There are magic weapons that provide damage type keywords (cold, fire, psychic, etc.), but that's about it.
There isn't any way to directly add the arcane keyword to your basic attacks. What you can do is get an at-will power that has the arcane keyword and that can be used as a basic attack. Unfortunately, while there are 3 main ways of doing that, none of them are particularly compatible with Bladesinger.
Option One: Be a Human and the Right Class
The original human racial feature (as opposed to the version Essentials humans get) allows you to choose an additional at-will power from your class (not your subclass). If you took this option you could choose any of the at-will powers available to any of the different wizard sub-classes (arcanist, sha'ir, mage, or witch). While this could be quite helpful to you in general, I don't believe any of those are melee powers that count as basic attacks (there may not even be any melee powers at all).
Option Two: Be a Paragon-Tier Half-Elf
The original half-elf racial feature lets you choose an at-will power from any other class and use it once per encounter. At level 11 you can then take a feat, Versatile Master, that then allows you to use that power at will. Notably, this allows you to take the one power that has the arcane keyword and counts as a melee basic attack: Eldritch Strike, from the warlock class. Unfortunately, that power depends on either Con or Cha, and is thus a poor choice for a Bladesinger (who depends on Int & Dex).
Option Three: Be a Hybrid
The hybrid rules let you essentially cram together two different classes to make a single class with 40% of class A and 40% of class B. This includes choosing a single at-will power for each class. You could hybrid your class with warlock to gain access to Eldritch Strike... except that no hybrid rules were ever published for Bladesinger to my knowledge, so it's impossible to hybrid Bladesinger with anything. On top of that, the hybrid rules are fairly underpowered; the vast majority of hybrid characters end up being strictly worse than a single-class character of either of the classes they combined.
As you can see, options 1 and 3 simply don't work for you, and option 2 would require a serious expenditure of resources, probably to the extent of making your character useless at a number of other things.
There is another way. If you can get a Wizard at will that is a melee attack (for example, Beast Switch), then the War Wizard of Cormyr Paragon Path allows you to turn that power into a basic attack. The main way of doing this would be by being a human and getting an extra at will, but the group that I am in allows the retraining of power granted by class features IF they have a level (such as Bladesinger's Magic Missile), so your mileage may vary.