Itinerary + last-day flight logistics (Osaka to Tokyo) need advice

Hey everyone! I could use some advice on a travel logistics question.

I already have a roundtrip flight booked from Home Airport → Tokyo (Haneda). But now I’m thinking of ending my trip in Osaka and booking a separate one-way flight from Osaka → Tokyo on my last day to catch my international flight home.

My questions:

If I book that Osaka → Tokyo flight separately, will the airline be able to check my luggage all the way through to my international flight, or will I have to pick it up and re-check it?

Will I need to go through security and immigration again in Tokyo before my flight back to the U.S.?

How much time would you recommend leaving between the two flights to be safe?

Is this even worth it, or would taking the Shinkansen back to Tokyo be a better option?

My current route is looking like:

Day 1: Tokyo

Day 2: Tokyo

Day 3: Tokyo

Day 4: Kamakura (day trip)

Day 5: Kawaguchiko (day trip)

Day 6: Tokyo

Day 7: Nagano City (shinkansen)

Day 8: Hakuba

Day 9: Hakuba

Day 10: (back to tokyo, friend is leaving)

Day 11: Kyoto (shinkansen)

Day 12: Kyoto

Day 13: Nara

Day 14: Osaka (friends leave)

Day 15: ??

Day 16: ?? and flight home from Haneda @ 8pm

Would really appreciate any advice or personal experiences, thank you!!

by Human_Baseball1625