I'm not saying it's not possible. I'm saying it's not possible with your budget. BER grows with speed unless you compensate with power or better alignment.
Yes, HeNe is helium neon laser.
I already linked you another laser. The one you select is simply underpowered. With 5mW and 1Gbps you have 5pJ per bit. Detector noise is 2fW/sqrt(Hz) with 1GHz bandwidth, it gives you 63pW of noise. That gives you theoretical SNR of 10**8. To get BER of 10**-12 you need SNR of 200. In theory, you have high margin. In practice, many things deteriorate ideal SNR.
Starting from transmission efficiency. If you focus it with typical M9 collimator (C105TMD-C, which btw is also not included in my price list) you'll get 3mm beam. With x20 expander I suggested you'll get 60mm wide beam. At this width divergence at the receiving end will be negligible, so assuming you get another beam expander you can compress it back to 3mm beam with little loss. Both beam expanders have transmittance of just 97%, giving you another 6% light loss. The detector has 0.8mm aperture, hence you'll only be able to detect 7% of original light unless you use additional optics. Pointing at something km away with mm accuracy is impossible without active stabilization, and here's probably the best place to add it. Around your main detector, you add 3/4 additional photodiode that detect which way the beam shifted and send it to the transmitter. This is a complicated system to build and program, so for starters it's probably a much better idea to spread the beam to a much wider cone and hope it hits the receiver somehow. This already reduces your SNR to around 6*10**6.
Next thing to tackle is atmosphere. At 2-4dB/km in slight haze, it attenuates your beam in roughly half. But more importantly, due to unevenness, atmosphere lensing spreads and distorts your beam considerably. In "professional" systems, this is compensated with adaptive optics beam forming, but here it's just a loss that has to be estimated and accommodated.
Another thing that reduces your SNR considerably is electronics. You need a wideband, low noise laser driver. This is a hard thing to achieve. You also need optimal receiver with adaptive threshold, low noise preamplifier, some filters to compensate pulse distortion and clock recovery system.
Till now, calculations only included thermal noise of the detector. Sunlight has a lot of power in NIR. To cope with that, you need a narrowband filter on the detector. The narrower the band of the filter, the better rejection of sun's interference. At the same time, the narrower the filter the higher losses it introduces and requires better and better stabilization of source wavelength. At some point in time, thermal stabilization of the diode may become important.
Obviously, you're not going to put just optical table outside, facing environment. You need some enclosure that'll pass 1550nm, hide system from straight light. The window is going to have some losses, but more importantly it'll get dusty inducing further losses.
In general, you can skip some steps by taking 1550nm fiber optics communication system and trying to modify it for FOS but it still requires a lot more knowledge than you seem to possess.
