I have to agree with most of Expat's advice here to which I'll add some of my own spin.
** Disclaimer - I work for a large US auto insurance but as a programmer. In that capacity I work daily with business analysts who all have come up from the line. So I know something (second hand) of how claims are "worked". **
When a claim is filed the company is required to set aside a certain amount of reserves to cover any potential settlement. This money then is not available for the company to invest and thus is no longer contributing to their bottom line. The amount they have to reserve will vary according to the potential liability. But even for claims with limited potential liability, such as this one, the amount the company is required to reserve will be high enough to "encourage" the company resolve it quickly.
So why is AIG not quickly resolving this claim? Well - insurance companies have to also watch costs very closely. As such they give their claims adjusters very limited authority in how much they can offer for settlement. Shockingly, a typical limit for a level 1 adjuster is just $500. I suspect that this claim, being that there are no injuries, is in the hands of such an adjuster.
Now from the companies perspective if they are obligated to pay $800 on this claim they would much rather pay it sooner rather than later. But this may not be how the claims adjuster sees it.
Claims adjusters are evaluated, in part, by how long, on average, claims assigned to them take to close. One way to quickly close a claim is to summarily deny it. But this is not practical when dealing with an adjuster from an opposing company as they can readily determine the merits of the claim for themselves. On the other hand the adjuster can simply pay each and every claim in full and close them quickly that way. This would make the company very unhappy (and unprofitable) very quickly so, in addition to monitoring average time to close they also monitor average pay out. If an adjuster is paying out more than what is expected then that calls for some in depth auditing of past settlements. Companies are quick to fire adjusters who are too generous. This is why they set strict limits on how much they can settle for.
So here is one, hypothetical, look at what is going on:
Insured calls in claim and largely admits being at fault. Statement is taken and as no injuries are reported and damage to the vehicle, if any, is likely minor and (really?) how much can a bike cost the claim is assigned to the very bottom level claim adjuster. This adjuster will have very little experience (probably less than 1 year maybe even less than 6 months) so they only have limited authority as to how much can pay - say $500. The adjuster instructs the insured to get estimates and begins assembling the claim file necessary to support any decision on the claim. From the insured's account the adjuster will know that a police report will be filed so that is ordered for review (note - I find it odd that, in this case, the adjuster is making the OP a settlement offer yet stating that they have not yet seen the police report. Standard practice is to review ALL the available documentation before making any offer).
Next the cyclist is contacted (or contacts them) and their statement is taken. The key thing here is for the insurance company to quickly gage the potential liability. Is the cyclist claiming any injury? No - good then it can stay with our level 1 adjuster. Once injuries are reported or there is other indication of higher potential liability the claim would have to be bumped up to a more experienced (and expensive) claims adjuster. The cyclist is told to send in their documented claims of damage. Time to sit back and have a cup of joe until the paperwork all arrives.
A few weeks later the adjuster gets the estimate from the cyclist. "What's this? $800 for a bike?" he shouts as coffee slurps onto his stain resistant dockers. "I can't pay that" he mutters to no one in particular. "What to do ... What to do.." he thinks. "I'm new in the job and the bosses are already on my back to close claims quicker. I don't want to go back to flipping burgers at McDonald's but I can only pay out $500." He could simply refer it up the line to the next higher level adjuster but then they would get credit for the closure not he.
"Hmmmm" he purrs as he reviews the claim file. The cyclist told him he didn't have a car hence the claim for bicycle rental. "What kind of person doesn't own a car? Even I own a car and Lord knows what a mess of financial trouble I'm in!"
"He must need money badly - I know $500 would be real good to me right now. Let's see the police report again to refresh my mind - oh! Not good here. Good thing I'm not dealing with another insurance company as we'd be dead to rights on this one. Our insured turned into the cyclists path and he couldn't stop. No moron would intentionally ram their unprotected body into the side of a 8000 pound SUV if it could be avoided. Well - maybe those morons on the "Best of Jacka$$" DVD I have at home would! Loved that show .... Better to pretend I haven't seen this yet".
So he decides to bluff his way - offer 60% ($480 - just under his limit) and pretend he hasn't seen he police report so he can, with as much a straight face as possible, maintain the cyclist is 40% at fault.
******* Here is what I would do **********
1) Recognize that, for whatever reason, the adjuster you are dealing has his hands tied and can't pay you the full amount of the claim.
2) Ask to speak with the adjuster's supervisor. Be professional and courteous. Part of your goal here is making them realize they are not dealing with a person of limited intellectual ability (after all in their minds anyone who doesn't own a car can't have great reasoning faculties). The supervisor also has authority to pay a larger amount. Ask the supervisor to please review your file, at their convenience, especially the police report listing their insured as at fault and give you a call back when they are ready to discuss it.
3) If they stick with the 60/40 determination (by the way this is something the adjuster made up out of thin air) because you "didn't avoid their insured's reckless maneuver" ask them if they, personally, would be willing to ram their body at 10, 15 or 20MPH into the side of an 8000# SUV?
4) Let them know, after a reasonable time period, if they haven't agreed to pay your claim that you will file suit in the local small claims court. I'd give them as much as 2 months or more on this. What you are doing here is saying "look - I know I'm entitled to the full amount and, what's more, I can wait for it". While the claim remains open the adjuster (and hopefully now the supervisor) will have to explain to their bosses why it hasn't been closed. While the claim remains open they have dead reserve not available for investing. They have already admitted partial liability so they can't deny the claim and close it that way.
5) Just as $800 isn't worth it to you to hire a lawyer - $800 is not worth it to them to defend a claim in a faraway court - especially with a police determination in your favor.
Good luck!