I say go for it, but know that you have no chance of doing it in 2 months. A couple years, maybe, if you have it in you, but not two months.
I had a similar story and background when I was freshman back in 1987 at UCLA. I also had only 2 years of running in high school, and with bad coaching at that, such that I would take 2 weeks to recover from a single hard workout because I ran them too hard, and wasn't taught to rein that effort in. My PRs were only 4:46r, and 10:00/10:02, and ran XC, but never did 5k xc (distances were like 2.95, 2.86, etc..). The others posters are right - they really just don't want you on the team. The coach, Bob Larson gave me 30 seconds of his time that first year. "Did you run at least 4:30 or 9:30 in high school". "No". Then something like they don't let you even try to walk on without those high school times - even if could run those times that day as a freshman.
I ended up biking on the college team freshman year. However, I couldn't ride the next summer because my bike frame broke and it took months to get it warrantied by Trek, so I went back to running on my own. I did some quality sessions like hill repeats, but mostly just ran a lot of miles in singles, and didn't concern myself with racing. I had no vehicle other than a bike, there was no internet then, and I was shy, so I had no idea how to get into open races, or that they even existed. My only races were the ones on campus - intramurals. I won the intramural XC race 2 times over a bunch of redshirts on the team, and won the intramural 2 mile the one time I did it by over a full lap.
I built up from 11 miles a day in singles to 20 miles a day in singles by my Jr. year, and then backed off. Running was just kind of therapy for me, and how I enjoyed spending 2.5 hours of my afternoons - I loved exploring the far off trails in the area, and I could only get there by running there, thus all those 20 milers. I was doing occasional workouts, but not too much. Some point during my Jr. year, I started adding 6 mile tempos at the beginning of a weekly 12-16 miler. The first one, I just wanted to go sub-6 pace, and I did. Within weeks I was doing the 6 miles in 32-31 minutes, and within a couple of months or so 30-31 minutes. My mileage was about 90-95 per week. I talked to the coach one more time at the end of my Jr. year, and he gave me about one minute of his time. He said that since I had only one year left before I graduated, it wasn't worth his time to even look at me... unless I ran an NCAA qualifying time. That would have been 29:10 for the 10,000, I think. The funny thing was that there was no one on the team that year that could hit that time. That was the last I talked to him. Later in the fall of my senior year, I was running my 6 milers under 30 minutes 3 times. I think n 29:46, 29:28, 29:46. These were solo, on the road, in trainers, without a warm up. I pretty sure I could have run about the same time on the track for 10,000 m in a paced train of runners wearing flats or spikes. So I got damn fast, but never got the college running experience that I wanted. That's life, I guess.
A year later, post undergrad graduation, I decided to try to put some real times down in races, so I started training again with goals. Unfortunately, I still didn't really understand training well and made the mistake of trying to follow a canned plan for a sub-30 10K (Self Coached Runner by Al Lawrence and Mark Sheid) without making any adjustments. The amount of faster paced running involved compared to what worked for me previously and trying to go prescribed paces instead of letting the running/speed come to me destroyed me about 3/4 of the way through the schedule, though I was able to do everything up to that point. I ended up getting chronic fatigue syndrome, or what they now call overtraining syndrome, and that lasted for 2 full years. Afterwards, I was never quite the same, though with half assed training, I ran 25:07 for 5 miles, and less than half assed training, a lot of 32 minute 10Ks.