I agree with abmj's rec to cut down the trip.
If your budget is only $3000, and I presume they'd like to avoid hosteling on their honeymoon, then it's too ambitious to plan a 3-week trip in Jpn.
If you really want to rough it, sure anything is possible.
But I think it's important for them to enjoy the trip, have security of reliable accommodations, and try good-looking food when they see it instead of having to pinch pennies all throughout the trip.

In a 14-day trip, if they really want to do Nagasaki to Sapporo and back down to Tokyo, they can actually do it. I wouldn't recommend that type of trip to some or probably most. However, if they're young and want to take in as much of Jpn as possible, they can do it. I've certainly done that in the past in Jpn and Europe, and those were some of the more memorable trips of mine.

I would get the 14-day Japan Rail pass.
* Try to fly into Nagasaki (or FUK), out of TYO. Open-jaw.
* If can't find good fare for above, then round-trip to TYO. Forget adding the 10,000yen foreigner airfare for TYO-Nagasaki portion, since saving $ is of the essence for this couple.

Assuming, flying into & out of TYO:
day 0: stay near Tokyo.
day 1: activate JR Pass; Tokyo to Nagasaki; stay in Nagasaki
day 2-4: Nagasaki area (some good day trips in the area); stay in Nagasaki
day 5: travel to Hiroshima; see Hiroshima and stay there.
day 6: Miyajima, then onto Kyoto; stay in/near Kyoto.
day 7: another day in Kyoto; stay in/near Kyoto.
day 8: Kyoto until mid/late afternoon; travel to Sendai; stay in Sendai.
day 9: travel to Hakodate; stay in Hakodate.
day 8: Hakodate until early afternoon; travel to Sapporo; stay in Sapporo.
day 9-10: Sapporo; stay in Sapporo.
day 11: Sapporo-Tokyo; stay near Tokyo.
day 12-13: Tokyo.
day 14: fly out.

Budget:
JR Pass x2: 93,000
Breakfast: included w/ hotel
Dinner: 24,000 daily
Snacks/lunch: 32,000 daily
Attractions/misc: 3,000 daily
Hotel: 9,500 daily (in places like Nagasaki and Sendai, can be ~7000; but in Tokyo and Hakodate, expect ~12,000; For Kyoto, I'd forget about staying in Kyoto but rather somewhere nearby.)

For a 14-day trip, that comes out to 346,000 yen. Just a hair over $3000. Is that doable for them?

Some might say you don't have to pay so much for meals or hotels in Jpn. Well, yes if you're just doing Yoshinoya and food court and konbini bentos, you can probably get by for as little as 2,000 daily for 2 on top of the free hotel breakfast. But then usually you will come across at least a half-dozen small bite type things or awesome drinks/dessert that could cost anywhere from 200~1000yen which you feel like you must try. Are you going to deny yourself the chance every time? Likewise with hotels, can you be happy staying in a 10m2 cubbyhole room at Toyoko every night?

I'd avoid April since it may still be too cold up in Hokkaido. Definitely avoid Golden Week.
Maybe May (before tsuyu season) or mid-late Oct (around the time typhoons become less frequent) might be the best.