Your child is already have problems, and has presumably realized it. Denying a diagnosis does not deny the problems. And a label is not necessarily bad. It's what you do with it that counts. By giving him a diagnosis, you are saying 'it's not you, you're not stupid, you're making avery effort, but your brain works differently, learns differently, has different strengths...' You are giving him the tools to get around his weaknesses and find his strengths. Information is power. Without information, how will you know how to help him or lobby for the extra support he needs?
He especially needs to know that his sister is not better than him, just different. She may be better at some things, and he will be better at others. Dyslexic brains are powerful. Don't see dyslexia as nothing but a problem - it causes problems with learning, but it also generates strengths in different areas.
My DD was diagnosed dyslexic after a dreadful first year in Primary (age 6 - we are not in the UK) when her confidence plummeted. Having a diagnosis enabled us to explain to her that what she was going to was not her fault, and to get her help and support in school and out. It enabled her to take control again, come back at friends who asked why she was so slow at some things. Now, (age 8) she knows she's bright, she works hard, and she's making excellent progress. She couldn't have done it without the understanding that came with diagnosis, and the support she has had since then.