I think we need to keep in mind that the GI are written for someone who says most of the LOTH every day.
I can’t personally see any harm in saying the Invitatory if it’s the only Office you do in the day. But a strict reading of the GI suggests that if there is no MP afterwards, you don’t say the Invitatory. Personally, if it were the only Office I prayed during the day, and I said it early in the morning, I would say the Invitatory, but not if I said it later in the morning, after noon or in the evening.
I think we can read between the lines and infer that the Invitatory is meant to be a early morning prayer, as the GI says it is to be said before MP, or before the OOR of the latter is said before MP. That is a very strong suggestion that the Invitatory is meant to be a morning rite and is not appropriate for the afternoon. Personally, I wouldn’t say it if I was saying the OOR any time after MP would normally be said, that is up to just before the hour of Terce. My own sense is that MP should be said at no later than 8:30 am, or in extenuating circumstances, maybe 9 am. But it’s not a hard-and-fast rule, just a rule-of-thumb that I use in my own prayer routine; I generally try to be structured and say the Offices at the same hour every day with some exceptions.
I normally say MP at 7:30 am (same time as at the abbey) in the winter months (except 6:30 on Saturdays as I have to be somewhere at 7:30 am), and at 6 am in the summer, immediately after the OOR. I use the same definition of summer as the abbey, that is from the Monday following the Octave of Easter until the feast of the Holy Cross on Sept. 14th (St. Benedict defined it as until Nov. 1st, and having been to Italy many times in early November, there is a distinct and fairly sudden season change in the 1st week; by November 1st in Quebec, summer is but a distant memory!)