Yes, it would be balanced to allow Enlarge/Reduce to double its effect when upcast.
Tons of spells have no "higher spell slots" effects, so there is no absolutely important need to add such a functionality here. However, since you asked:
+1d4 dg is basically +2.5 dg (on average).
+2d4 dg is basically +5 dg (on average).
Combat wise, the first size increase grants Advantage to Str checks and Str saves, plus the extra dg, while your second size increase only gives the extra dg and no more Str.
Can be "reasoned" that at that size all your new extra strength is needed just to hold yourself up, albeit that seems a bit anti-climatic. Exact same logic for Reduce. You just reach some smallness limit, and aren't all that weaker than you look, maybe you'd a bit denser whatever.
Compare with Polymorph, which is "only" a 4th level spell. Polymorph allows one to take the "almost humanoid" shape of a Giant Ape: a huge beast with 23 Str, dealing 7d6+6 dg.
Compare also with Vampiric Touch: Level 3, +3d6 dg, and you heal up half of that!
So, yeah, putting the 2nd size increase of Enlarge / Reduce spell at level 7 is putting it at a way too high spell slot level.
Also, the weight increase can be addressed by supposing that past the first size increase, mass doesn't increase as much, "explaining" that only the extra dg is gained.
Finally, True Polymorph is level 9 and can do way more than just give you Tiny or Gargantuan size. So, an Enlarge / Reduce variant allowing to reach those extreme sizes should be level 8 at most.
So, here is a slightly better balanced version:
Always: Str check and Str saves Advantage (Enlarge) or Disadvantage (Reduce).
Enlarge - Spell Slot level used:
2-4: +1 size, x2 height, x8 mass, +1d4 dg.
5-7: +2 size, x4 height, x32 mass, +2d4 dg.
8-9: +3 size, x8 height, x128 mass, +3d4 dg.
Size increase (and thus height and mass changes) limited to Gargantuan size. However, the dg modifier remains even if so limited (you're just slightly bigger but with clearly more enlarged muscles).
Reduce - Spell Slot level used:
2-4: -1 size, 1/2 height, 1/8 mass, -1d4 dg (minimum 1).
5-7: -2 size, 1/4 height, 1/32 mass, -2d4 dg (minimum 1).
8-9: -3 size, 1/8 height, 1/128 mass, -3d4 dg (minimum 1).
Size decrease (and thus height and mass changes) limited to Tiny size. However, the dg modifier remains even if so limited (you're just slightly smaller but with clearly more reduced muscles).
Weight analysis: Bob the barbarian weighting at 200 lb + 100 of equipment, gets Enlarged.
Large Bob: 2400 lb = about 1 ton. Most structures will have no problem supporting that but wooden floors might creak a bit. Note that if Bob is inside an inn or something, he won't even be "allowed" to Reach Large size at all, growing only enough for his head to touch the ceiling.
Huge Bob: 9600 lb = about 4 tons. Wooden floors will definitely creak loudly, thin ones may even break. Stone structures have zero problem.
Gargantuan Bob: 38400 lb = about 16 tons. Wooden floors break under the weight unless they are quite thick or sturdy, and stone remains ok, unless it is a quite weak stone structure.
Other Ways To Improve Effect With Upcasting
Another way to make a spell work using higher level slots without tweaking the intent and result of the spell too much, is to make higher spell slots affect either duration or number of targets instead.
Edit: In fact after reading comments and a bit of thought, I did not realize how much of a good Reach advantage this could give. Personally, I'd just tell the player: nah, can't do. Tons of other spells don't have up-casting, even if we could find all kinds of ways to turn lots of these spells into up-castable ones, it would just be a big hassle and in the end just give even more versatility, and thus power, to a class that already ends up so powerful it ends up overshadowing almost everybody else. The fact that many sells aren't up-castable is built-in, so, no, end of story.
"Now, if you wanted to do spell reasearch on it to try to improve the exisrting spell, spending months and tons of gold? That's another story."
Then I'd let the player do the "research". I'd probably decide that some other NPC, centuries ago, already tried doing the exact same thing (because it is such an obvious "improvement attempt", and of course since the spell doesn't allow it per RAW, that duce obviously failed. Then I'd determine the odds of the player succeeding himself. Throwing tons more time and money at it would not make the odds better. In the end the player could probably create a completely separate 3rd level spell that allows him to become huge size, and fall to make a generic super-enlarge-super-reduce 1st level spell.