Crystal Skull (2008) was set in 1957, so Indy 5 ('19) should be set around 1968-69.
Cook up some kind of intrigue with the moon landing.
I'd recast "Mutt" with either of two kinda-similar looking guys: Orlando Bloom or Milo Ventmiglia.
Neither is a big huge rough tough he-man type so recasting from LaBeouf won't be visually jarring (i.e, Chris Hemsworth).
They're also both late 30s so they could fudge their age a bit and introduce a mid-teens "Henry IV" they could set up for a full action role as an 18-20 year-old in Indy 6 in about 2025.
When that reboot comes, maybe you could pull a "Godfather II"dual-timeline deal and cast two very similar-looking 20-ish guys -- one as Indy in the "Roaring Twenties" prequel timeline, and another as "Henry IV" as the "Generation Jones" grandson forging ahead in the late 1970s.