I personally think you should switch the order of your adjectives to rich and handsome. There's almost always an order to descriptive words that sounds more natural. I think 99 times out of 100 you'll hear “rich and handsome.”
I believe the order of adjective in order to sound more natural (traditional) is to list them like this...
1. Non-Physical Perceived Traits (Opinion)
2. Real Physical Traits
The cute little cat ✔️ The little cute cat ❌
The mean old hag✔️ The old mean hag❌
The rude obese man✔️ The obese rude man ❌
There are probably many exceptions to this rule also. English is a language full of single exceptions to a general rule. Honestly, in my opinion the order of the adjectives listed really is not very important. It really doesn't change the meaning of the sentence. Sometimes, native speakers will purposely say adjectives in a strange order especially on writing in order to add variety to the sounds of their sentences. A lot of the so called "correct" adjective order comes from listening and repeating other native speakers. The listing of adjectives in any order also does not affect the grammar of your sentence! "He doesn't have a girlfriend even though he is handsome and rich." sounds just fine to my ears! 😄 If anything, it sounds unique and flavorful which is my favorite part about using language! I like people who use language in new and interesting ways as long as meaning is kept intact. Because the meaning of your sentence is clear and grammatically correct, I think it is natural. Even I would probably be saying "handsome and rich" from time-to-time so that my sentences maintain variety. 😄
I agree with slyfin. For the three examples, the correct ones all sound more natural to me. In our example here, I think it matters less and, like slyfin said, might emphasize the description even more.