It's up to the DM
I think you've outlined what would be the strictest rules-as-written case in favor of allowing this. You have spell slots, it's on (one of) your spell lists, so you can choose it. However, I think there is a strong case to be made against this ruling based on how the rules treat multiclassed spellcasters. For this reason, I'm saying the answer here is "up to the DM" because you should not be surprised if your DM does not allow this.
In the rules for multiclassed spellcasters, we see:
Spells Known and Prepared. You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class.
Each spell you know and prepare is associated with one of your classes, and you use the spellcasting ability of that class when you cast the spell. Similarly, a spellcasting focus, such as a holy symbol, can be used only for the spells from the class associated with that focus.
These rules rather unambiguously take the stance of "pretend you are single classed when figuring out how you spellcasting features work". So while it is not clear to me that these rules explicitly forbid the combo you ask about in the question, using these rules as heuristic for ruling on other features does.
Further, I'm inclined to rule against this combo on the basis of balance alone, rules notwithstanding. That is, this combo is extremely powerful. You mean all I need is one level in cleric, and I can cast any spell on the cleric spell list I have slot levels for? This drastically boosts the flexibility of any spellcaster, especially wizards who generally don't have access to healing spells.
In summary, yes, there is probably a strict rules-as-written case to be made here, but you should not be surprised by a DM disregarding any RAW argument for this.