I hear you. I go through periods when I just don't have the time to train for even 10 minutes, kids, work, study, chores, etc. I've tried getting up a little earlier and getting in some swings and TGUs which worked ok.
Something to consider is just using this period to rest your body and rather than make progress, just prevent regress so you can put the hammer down when time frees up.
There are some good barbell complexes that can be done very quickly as a circuit; think "progressive pulls" Marty Gallagher describes. Another would be MTN tactical "barbell complex". Complexes could be done in a variety of frequency/volume scenarios.
Shortest Daily Sessions:
Daily dose deadlifts (1 set daily) all week + high volume presses on the weekend. Because you don't have micro-plates, you'll have to start over each week adding 1 rep each week. Not sure when exactly to switch to the next plate though, maybe when your week starts at 8 reps.
For an anti-thesis to short daily sessions:
With only training 1-2 consecutive days a week, keep your volume high those days and expect to be sore the next few days. I'm thinking something pretty brutal like german volume training (10x10 @ ~60% 1RM)
If you could get one session in during the week, and another on the weekend, you could do a little less volume at a little higher weight. I'm thinking something more like 5x5 @ ~80% 1RM.
Each session, I would pick one movement from each of the major planes: press, pull, hinge, squat. If you have time, maybe add some accessory movements after. Again, without micro plates, you'll have to go through the programs adding volume until you can start over at a low volume with a higher weight.