Let's assume that the coalition loses badly enough that it cannot cobble together a working majority in the Upper House. The Diet convenes to let the Upper House elect its officers (including, presumably, a new president) and promptly goes into summer recess. At some point in September or thereabouts, the Diet convenes again, this time to consider an emergency budget to be submitted by the Cabinet. (No fiscal year has gone by without an emergency budget being submitted and passed within my memory.) The budget plan that the administration submits will include relief and reconstruction funds for the earthquake victims, so it will be difficult for the opposition to resist. In any case, the budget can be passed without the consent of the Upper House, so, at worst, it's some bad press. Unless there are pressing legislative concerns, the Diet can and will recess right then and there.
The same Constitutional rule applies to the main budget, which will be introduced at the next regular Diet session (usually convened in January), sometime in March. However, much new legislation will be needed to implement the budget, and that is where an Upper House deadlock will hurt. Yes, the coalition does have the Lower House supermajority to overrule the Upper House. But the Japanese media traditionally does not look kindly on a majority acting unilaterally against an opposition that has dug its heels in. More importantly, an opposition controlling procedures in the Upper House can make it extremely difficult for meaningful legislation to make it through the two Houses and back to the Lower House again, unless you are willing to extend the Diet session indefinitely. I don't think that the media will look kindly on such a prospect either.
Thus, at some point in early spring, the situation will likely become untenable for the administration, forcing the prime minister to dissolve the Lower House for a general election.
This is a scenario that can unfold regardless of who the prime minister is. So the question is, is the LDP better off with or without Mr. Abe in that situation? There is no obvious LDP leader that could run this near-caretaker administration and fight the general election better than Mr. Abe. So, it will be Mr. Abe's job to give up, if he so chooses.
Do I believe in what I'm saying? It is speculative in spots, but it is at least plausible. So there you are.