Wishing someone good luck in continuing something without yourself without sounding hypocritical or sarcastic

I would suggest that your "Wishing you all the best with the project" line works just fine, unless you and the customer have a history of sarcasm.

'I’m really sorry we didn’t get your business this time as I think we could have worked well together. Nevertheless, I do hope your project goes well, and I wish you every success with your chosen partner.'

How about something like ...

Mr. Smith,

Thank you for the opportunity to present our services to you. I appreciate your time and attention, and am grateful to have met you.

I hope we have an opportunity to work together in the future. May your project surpass all of your expectations. If I can serve you in any way, please, don't hesitate to contact me.

Sincerely

This approach wishes well, shows gratitude, while also putting you in mind for future opportunities.