“To spend (money) on doing something” is absolutely correct (and probably the simplest, most common way to word it). For example, you could spend $10 on getting your car washed.

Both "spending something ON/IN doing something" are correct. Example: "I spent $500 on/in renovating my house". Where I am, the 'on' version is more common and using 'in' sounds very casual. If you drop the 'doing something', then you should use 'on', not 'in'. For example, "I spent $500 on my house". This is implying you spent $500 on doing something (maintenance ?) to your house. This work could be inside or outside. If had used 'in' and said "I spent $500 in my house", the meaning changes as you are implying you only did something within your house - not outside.