Based on the amount of points you have, you should have close to enough miles to fly Business class both ways, using UA miles on the outbound and DL miles on the return (or vice versa).
It looks like UA partner flights are 88K per person each way (would only be 80K if you can find UA metal, but I didn't see any UA saver space in a quick cursory search out of NYC). That's 176K for two people, which isn't too far off from what you currently have. If you have any Chase UR points or Marriott points, you can top off your account and then book them that way. You could buy the points difference if you don't have either of the above, which unfortunately would be a sunk cash cost if you have to cancel the trip. Or you could book your son in J and you in Y to save miles and cash, and ensure he gets a great experience even if it's at your expense.
On the return, the days that Korean (KE) flights are available using skymiles, DL is charging 85K per person, which is not a bad price. The KE product is excellent.
Since the availability on partners to TYO is pretty widespread in June as of now, it may be smart to think of a strategy to quickly earn about 30K united miles with minimal cash outlay. Perhaps it's worth looking into signing up for a UA, Chase, or Marriott credit card depending on what bonus you can get and what minimum spend threshold is most realistic for your current situation.