Have a look at NCL.
Many or even all the Princess cruises do anytime dining. We are travelling from Southampton next summer on the Grand and have opted for anytime dining. We have travelled with princess many time and find this dining arrangement works well. Check the princess web site or even the brochures. Royal caribbean dosnt do anytime dining so do not waste eyour time checking with them.
P & O cruises offer a Freedom Dining option as we have been on the Oceana twice and used this.