A recent article in Audio Masterclass described the problem of miking a clarinet. In short, the sound doesn't only come from the bell, it comes from the whole length of the instrument. So a mic taped to the bell won't give a realistic sound
The article did propose a solution, but it turns out there is a solution ready-made, and that is the AMT (Applied Microphone Technology) WS. As their literature says...
“The WS is a low profile double condenser microphone that attaches to the clarinet, alto and bass flute, or oboe with a ultra-miniature low mass clamp. It is specifically designed for the timbre of the clarinet. The microphone provides pure tonal reproduction while being unobtrusive. It has an even reproduction of sound across the entire instrument. The AMT WS features a flexible gooseneck, allowing the player to position the microphone for the best possible tone reproduction. Its rugged construction of anodized aviation aluminum makes it highly durable, while being lightweight and non-corrosive. The WS Features one microphone at the bell and second positioned over the keys to provide a truer reproduction for the instrument's tone quality.”
In fact, AMT has a whole range of clip-on microphones for various instruments. Well worth looking into, I would say.
Great home recording starts with a great home recording studio. It doesn't need to be expensive if you know how to select the right equipment for your needs.