You should be able to interact the gate and enter after you get the blue rabbit. And dont know how this dialogue bug happened with you. Never faced these bugs, i have played the game multiple times. This might be an issue with the browser version, try downloading and play and see if the bugs still persists and let me know. Also you can check the gameplay video in the comments someone played, at the end there is bug and for that i mentioned the solution at the top of itch page, this will help. Thankyou for your time. :)