Could we use particle colliders as fusion generators?

A subtle problem you seem to overlook is that the proton-proton cross section is very small, about 0.07 barns (a barn is $10^{-28}$ square meters) at the LHC energies and not dramatically different at your lower "fusion energies". It means that at the LHC, much like at your dream machine, most of the protons simply don't hit their partners. It is not really possible to focus the proton beams arbitrarily accurately, for various reasons (the uncertainty principle is the truly unavoidable effect: you either localize the beams in the transverse direction, into a "thin pipe", or you specify that the velocity in this transverse direction is zero which is needed if you want to preserve the location "in the thin pipe" in the future, but you can't do both at the same moment). If it were possible, the LHC would be among the first ones that would use the method, to increase the luminosity.

So if you accelerate two beams of protons against each other, an overwhelming majority of them will simply continue in their original motion. (The protons in the LHC have to orbit for half an hour or so – tens of millions of revolutions – before one-half of them collides or disappears.) It costs some energy to accelerate the protons to these energies and you want this energy to be returned from fusion, with some bonus. But the fusion only returns you the energy from the protons that collided (some of them could create helium at your energies but there will always be nonzero probabilities of other final states; it's not a deterministic system that always produces the same final state for a given initial state; quantum mechanics says that the outcomes are random) which is a tiny portion of the protons. So you will be losing most of the energy you invested for the acceleration. Note that the LHC consumes as much energy as the households in Geneva combined and it just produces collisions of protons whose energy is smaller than a joule per pair.

To increase the fraction of the protons that hit their partners, you either need to send them to the collision course repeatedly, like at the LHC, but then you need to pump extra energy to the protons that they lose by the synchrotron radiation (which is always nonzero if the acceleration vector is nonzero, e.g. for all circular paths). Or you will need to dramatically increase the density of the beams.

But if there are many protons in the beam, they will electrically repel each other and you will become unable to focus them for collisions, too. So what you need to do is to electrically neutralize the high-density proton beam and then you have nothing else than the plasma and you face the usual tokamak problems how to stabilize it. Note that the electrons respond totally differently to the external electromagnetic fields than the protons. The LHC uses both electric and magnetic fields to accelerate the protons but to keep the plasma neutral, you must avoid electric fields.

Tokamaks only work with magnetic fields. Whether they will ever become fully working and feasible remains to be seen but the absence of the electric fields implies that they don't have much in common with particle accelerators.

Actually, this has been done, but it's not sustainable. Wikipedia has a brief explanation:

Accelerator-based light-ion fusion is a technique using particle accelerators to achieve particle kinetic energies sufficient to induce light-ion fusion reactions. Accelerating light ions is relatively easy, and can be done in an efficient manner—all it takes is a vacuum tube, a pair of electrodes, and a high-voltage transformer; fusion can be observed with as little as 10 kV between electrodes. The key problem with accelerator-based fusion (and with cold targets in general) is that fusion cross sections are many orders of magnitude lower than Coulomb interaction cross sections. Therefore the vast majority of ions end up expending their energy on bremsstrahlung and ionization of atoms of the target.

Basically, a particle accelerator provides many ways for the accelerating particles to lose energy that are unrelated to the actual collision.

Take the LHC as an example. It consumes over 200 MW of electricity, and generates up to 600 million collisions per second. If the particles being accelerated were deuterium and tritium nuclei, each fusion event would release about 17 MeV of energy. So if somehow every single one of those 600 million collisions per second was a successful D-T fusion event, the total power generated through fusion would be

$$\frac{17\text{ MeV}}{\text{collision}}\times \frac{6\times 10^8\text{ collisions}}{\mathrm{s}} = 1.6\text{ mW}$$

for an efficiency of 0.0000001%. That's not very good at all. A sustained nuclear fusion reactor needs to have an efficiency of greater than 100%.

Granted, most of the energy consumed by the LHC goes into giving the protons or ions kinetic energy, which isn't necessary for fusion. If you subtract off the 8 TeV of energy per proton, that could raise the efficiency by a factor of about a million (bearing in mind that there are still many reasons that what I'm talking about is impossible), but it'd still only be around 0.1%.