By the look of it, I have a made conjecture, it could be reason he doesn't want to go there alone. If Miwako's parents had ask her before to find a husband, it's likely... Takagi has no wish to go visit his parents without girlfriend as he can predict when visiting parents just like going to interrogation room hearing 'do you have a girlfriend?' 'When you will marry?'
Takagi doesn't have much luck when propose to her last time and Miwako wear the ring like she think it's a necklace. If out of ideas and don't know how to say... guys find hard how to break or propose and in the end eventually he didn't. Deeply in Takagi's heart, he wish one time he did succeed in one go instead of losing ring in the sea or being mistaken for giving the ring.