Of course, you're taking us to the King-Byng debate and the issue of whether a Governor General can call on the second party in the House when the party with the most seats reports that it does not enjoy the confidence of the House.
My view is that Governor General Byng was absolutely right and that Prime Minister King was absolutely wrong. It was constitutionally perfectly proper for the Governor General to call on the leader of the Conservative Party to seek to form a government.
That was a political misjudgment, because in fact Mr. Meighen could not command a majority, as became evident within some 48 hours. But constitutionally that was the right call.
Now--as I understand what you're asking--does the Governor General's right to do extinguish over time during the life of a Parliament? I would argue that as a constitutional matter...bearing in mind that we're talking about political conventions here, but to the extent they are unwrittenly constitutional in the British tradition. The Governor General always has the power to call on another parliamentarian to try to form a government. The issue is whether that will work politically when the next election comes along. The closer the next election is, the riskier it is for all concerned. But that's politics, not constitution.