If you handload, there are a quite a few good wildcats that perform amazingly in the Contender and will get you 1800 fps at well over 300 yards, which is all you need for any game, including elk.
The other thing that is good is that a lot of the bullets that are a bit too "splashy" at shorter distances in a rifle work wonderfully at the more moderate short range velocities of the Contender. The Hornady V-Max being one of the prime examples.
While your 35 Remington will work, you are very limited in bullet selection and in order to get the best performance out of it, you will end up with some pretty stout recoil. Just like with a rifle, proper follow-through is a critical component to consistent accuracy with hunting handguns and heavy recoil makes that extremely difficult. It takes a lot of practice and trust me, a long range session with full-power 35 Remington loads will not be fun.
So, here is a list of cartridges that I would recommend based on your stated desires. All are reasonably available in 14" barrels, and die and components should be easy to source as well.
6TCU, 6.5TCU, or 7TCU - Uses 223 brass (however if you decide on 6TCU, I recommend necking up 204 Ruger cases and trimming them back down so you can get proper neck lengths unless you find a source of 222 Rem Mag brass). Bullet selection is great for all of these. With a 14" barrel, using bullets in the 120-130 range for the 7mm and 6.5mm and 95-105 for the 6mm, MVs in the 2300-2400 are very achievable. That puts your 1800 fps velocity threshold between 250-325 yards, depending on the bullet and actual MV.
7-30 Waters - Some of the same justifications as the 7TCU. Readily available, economical brass (lots of good 30-30 brass out there). The main difference comes from the fact that you can get those high 2300s up to high 2400s with 130-140 weight bullets in the 14" barrel. If you stay with the 120 class bullets you can push up over 2500 fps, which extends your range to around 400 yards. With my 13" barrel, I am getting 2577 with a 120 grain NBT. That has me crossing the 1800 fps threshold at 394 yards. I would not hesitate to shoot an elk with that setup.
30-30AI - Yes the 30-30 will work, but the AI version really shines in the single shots and this is one cartridge that really sees improvement with the AI version. With the 125 grain NBT or Hornady SST, you can push 2400 fps in a 14" barrel. The Sierra 125 pro-hunter is another good option. With those your 1800 fps threshold is going to be in the neighborhood of 250 yards. However as with all of these, if you go with a lighter construction on the bullet (such as some of the lighter 30 cal "Varmint" bullets) you can push that threshold down to 1500-1600 fps which would get you another 50-100 yards.
30 Herrett - Another oldie but goodie. Uses 30-30 brass, but the forming process is a bit more involved that just loading and shooting like with the 30-30AI. Velocities with it are going to be about 100-150 fps slower than with the 30-30AI. Still a solid performer to over 200 yards with the right bullets.
Standard 30-30 - As was said in another post, you are not limited to flat or flex-nosed bullets in the Contender so you can load better bullets, however with the availability of the AI and the Herrett, both of which perform better than the standard 30-30, I put this last on the list. The only reason I would go with this over the AI or Herrett would be if you want to shoot factory ammo.
There are other really good and fun cartridges that can be shot in the Contender (the 6.5JDJ for example), however the availability of dies and components (namely the parent brass) just isn't there. Additionally, I didn't include any of the standard pistol cartridges because, in my opinion, unless you plan on treestand or blind hunting, they just aren't that viable as serious hunting cartridges for out West.
