It is only by, well, a shocking degree of ignorance that the the Book of Acts as we know it contains this characterization of what Paul said.
Paul studied under Gamaliel, believed to have been the preeminent Jewish scholar of his day, and not possibly could have said what is attributed to him at Acts 13:16 ff.
[Edit: added verse references to the next two paragraphs.]
The Judahite line, leading TO David, is something Paul would have known by heart, well, just as many of us do. That line is Judah → Pharez → Hezron → Ram → Amminidab → Nahshon → Salma → Boaz → Obed → Jesse → David (Gen 46:12; 1 Chron 2:9-12, 15b, Ruth 4.18-22).
Too he would have known that Nahshon, as captain over the children of Judah (Num 1:5a, 7), was present with Moses, leaving only his son Salma to enter Canaan (Num 26.65).
The way the passage in Acts is translated, not to mention whatever its author may have "believed" Paul said, presents us with a cartoon version of Paul, a version who wouldn't have even known it can't take 450-500 years to span merely 3-5 generations. The translation/author characterizes him as believing that even 450 years after Salma was old enough to have sons, David had still yet to be born. The average was 40-45 years per generation, so even though David was the child of an aging father, still we're looking at less than 200 years. Even Jacob saw three generations before migrating to Goshen at age 130. And even Abraham, who lived to age 175, saw at least two generations (Jacob was born when he was 160). So, again, the stuff of the passage you cite is just cartoonish.
[Edit: parsed and added verse references to the next three paragraphs.]
In citing a number like 450 --and I'm not saying he did not-- the only thing Paul can have been doing is adding a buffer zone to 430 for the time it took Joshua to subdue Canaan. That's 430 by one tradition (Gal 3.17) plus 6 years entailed of conquest (Num 10:11; 13:1-2, 6, 16; Josh 14:7-10), but since it's tantamount to 470 by another tradition (plus those same 6 years), all Paul would have been doing is splitting the difference as an estimate. It just depends on whether that 430 years was thought to have ended with Moses receiving the Law, or forty years later with Joshua forcing its ratification.
To reiterate, Paul knew that Judaism commenced with the birth of Isaac, and that the second covenant (the Law) stood as the cap for the famed 430-year span. But while one camp (in a split akin to Sadducees versus Pharisees) believed the 430-year span ended at the GIVING of the Law, another believed it capped off forty years later only when ratified —see circumcision at Gilgal (Josh 5:2ff). We see this difference of opinion reflected in the fact that Septuagint authors took the liberty to say 440 instead of 480 at 1 Kgs 6:1. Paul really gets into this in the third chapter of Galatians.
The point is, no, somewhere between translation and authorship the statements you cite, as they're laid out in the Book of Acts, are useless outside the framework of already HAVING a bead on what they're imagined to support. It can be adduced that the span from the Exodus to the death of king David is just under 300 years, though I won't get into all of that here. And forget trying to apply Jephthah's reference to 300 years "mid judges" (Jdgs 11:1ff) or, that is, using that as evidence for a longer period. The offspring of Isaac, even starting with Esau (Gen 32:3, 36:1ff), were settling deep in the southern Levant, and I'm confident that when Joseph and his large entourage deliberately passed through that same region (Gen 50:10-11) to let that branch of the family share in mourning Jacob, there were many opportunities for members of his entourage to peel off and settle in Canaan anew. That's just what all the literary evidence points to. Even by the time of Moses and Joshua, after all, there were an awful lot of non-Hebrews documented as part of the various lineages, and even Caleb himself is famously the son of a Kennizite (Num 32:12, Josh 14:6, 14).
