Need to identify it before going too much further. The leaves look like a bit like Illawarra flame tree (Brachychiton acerifolius) - do you know?
The tree is lopsided partly because of those palms "encroaching" on it. Are they on the street or in a neighbouring yard?
The lopsidedness won't necessarily make it fall - please respond with what you know about the tree's identity and approximate age, whether you planted or inherited it with the property.

Thanks for your response! I didn't plant it and I have no idea what it is. Yes, it was inherited when I bought the house. It loses leaves every second year, so I was inferring it was some sort of maple? No idea really.

Without knowing what it is, it's impossible to say how its growth would react to a "topping". I really would suggest you take some photos and leaves to a reputable nursery or arborist for ID.
My fear is that, if it's a species which naturally grows with a single trunk, then topping it is going to do it damage and if it survives and re-shoots it will do so with potentially weak side-growth. Also, the topping may not cause any re-growth below the cut so you are then stuck with the problem of no lower branches and a lot of sappy upper growth.
It is also difficult to assess, from the photo, just how close to the wall it is. Either my guess or yours, has the potential to grow into a VERY big tree that could do structural damage.
Whatever it is, it does not like the competition for light from those neighbouring palms.
I know you appreciate the privacy and it looks very pretty with the string lights (?) but I'd suggest replacing it before it becomes more of a problem. You can always buy an advanced specimen tree/shrub or even pre-grow one to take its place. Or use a temp screen/vine while you are preparing for a replacement