As per RAW - No, you can't use the spells in your spellbook as Artificer Spells
In general Artificer Spells, just like Wizard Spells or Warlock Spells are those spells that are in the list of available spells for the class.
There are exceptions to this rule. Certain class features might state that your list of available spells is bigger - for example Warlocks choose a patron that grants them certain spells that are not in the list of Warlock Spells that you would find at the end of the PHB. But: This feature states that these spells count as Warlock Spells for you.
The Wizard's spellbook for example allows you to transcribe Wizard Spells into the spellbook.
As such you can normally not infuse spells that are in your spellbook. You have to learn them through a class feature of your Artificer class that explicitly says that the spell counts as an Artificer Spell. Otherwise you would learn Wizard Spells when leveling your Wizard side, Artificer Spells when leveling your Artificer side and you could transcribe Wizard Spells into your spellbook - not Artificer Spells. Even if they are in both lists, they don't count for both at the same time.
Unearthed Arcana is playtesting material
Usual caveat: as you said yourself, Unearthed Arcana is not balanced for multiclassing and as such there may be problems when it comes to situations like these. Test the playtesting material and if you think something is too powerful/not powerful enough you should discuss with your DM how to adjust it.
Saying that every spell in the spellbook that is also in the list of Artificer Spells counts as an Artificer Spell for the purpose of class features from the Artificer when multiclassing is a possible solution. Or saying that you can transcribe spells for all classes where you have at least one level, so that you could treat the spell as an Artificer Spell.
But here you are in homebrew territory and this might have unintended side effects.