After careful consideration, I would have to say that Battletech would win on each front, especially if we are going ~3060s era. A small breakdown, shall we?
1) Spartans vs. Elementals: Elementals win hands down. Based on the research I am seeing, the spartans are supersoldiers who are given various forms of weaponry to combat their foes, up to and including anti-tank weaponry. Elementals can take up to and including a medium laser to the chest and still fight back. While I have no doubt that the Spartans would eventually pick up on the fact that their anti-infantry weaponry is doing them no good against the battle armor, I also question how quickly they would. When the Inner Sphere first encountered them, the elemental battle armor took out how many battlemechs (!) before the Kuritans figured out to use heavy weapons on them? Combined with the variability in terms of elemental battle armor (small laser, ******, heavy machine gun, AP gauss, etc.) leading to the a certain amount of unpredictability on the battlefield, the Elementals would likely take the Spartans down.
2) Vehicles on each side: Without the inclusion of Battlemechs, I would put this as very even (even if vehicles are difficult to control in my experience in HALO). While the Inner Sphere has more varieties of vehicle, the HALO universe would also likely have better soldiers inside them (imagine a Clan Elemental genotype for a tank, as that's essentially what Spartans are). Especially in combination with infantry assaults, I could see the USNC holding an advantage over the Inner Sphere.
HOWEVER! Then we include 16 meter walking titans that put all of the tanks to shame on both sides. Assuming the same vehicle rules are in effect (TT style =D), Battlemechs have a massive advantage over vehicles in their armor capacity, their ability to mount weaponry, and even their ability to maneuver over terrain (hovercraft non-inclusive). I could see an IS commander and a USNC commander duking it out, evenly matched, when the USNC commander says, "Holy [redacted], what is that?" as an Atlas walks over the battlefield and annihilates an M808B Main Battle Tank with an AC/20 and 3 Warthogs with its lasers. And this isn't even including Clan Tech. Substitute a Mad Dog (Vulture) or Timber Wolf (Mad Cat) for the same effect, only worse.
3) Space battles: Has anyone ever looked at the armaments for the two naval forces? A USNC destroyer has the BTech equivalent of roughly 2 NGauss, 3 Nuclear Missles (which, if fired, will cause the entire IS to stop fighting each other and start fighting the USNC), 2 sets of Naval Missles (IDK what size), and 8x AC10s. Very respectable.
http://www.sarna.net/wiki/Aegis
This is what the USNC is dealing with. In terms of raw firepower, the Aegis wins hands down. However, this could get messy due to the IS's consideration that Warships and their ilk are considered sacrosanct. An engagement in space initially would likely end poorly for the IS (especially with the consideration of the aforementioned nuclear warheads) until the IS finally said "[redacted] this!" and began using their weapons effectively. This gives the USNC a marginal advantage early game, but politically gives the IS a reason to stop bickering and nuke the WoBblies... er... USNC.
Verdict: Based on the numbers given I would have to say that the Inner Sphere would have the advantage over the USNC in most regards and, even in space, could compensate easily enough given political maneuverings of the IS. While the USNC is great for what it does, it lacks the raw firepower that Battletech brings to the field and, in the case of nuclear weapons, the IS could bring to bear as well. Now, please feel free to critique/flame.