Just doing a little research, it appears that most project lamps should last about 2000 hours of use (but noting that actual life falls on a bell curve and may be more or less than that). Oz Lamp indicates that with good care, a lamp should last 2000-4000 hours.
If one assumes that 2000 hours is an expected life, then if the lamp only lasted 4 months, one would have to use the projector about 16-17 hours each day to achieve the expected life span. I can't imagine a projector with high daily use as such being used in a residential environment (maybe in a commercial environment it would be more likely - e.g. shopping centre displays).
Even if the 7 months is taken as the life (say it is difficult to prove that the projector lamp was not installed on the day or near day of purchase), this would correspond to a 9-10 hour average daily use. This also seems high in a residential application, but not unrealistic for a commercial environment (e.g. offices etc).
Have you tried using the above information to demonstrate that a lamp should have lasted longer, assuming your own use isn't anywhere near that outlined above?
Also, is it possible for you to confirm that you took the necessary measures outlined on the Oz Lamp website to ensure that lamp life was maximised. If you can then this would negate one of their potential arguments that the lamp could have overheaded or subject to operating conditions which could have lead to its failure.
Also check the website of the manufacturer (and others with similar spec projectors) to see what warranty they offer on the lamps when the projector is new and has lamp installed. This will provide you with an idea of what manufacturers think is reasonable in relation to warranted life. Looking at Epson, some of their new projectors come with a 12 month/750 hour lamp warranty. 750 hours may be achievable in a residential environment for high user over 4 months (average 6-7 hours per day) or moderate user over 7 months (average 3-4 hours per day use).
At the end of the day, it appears that projector lamps have a reasonably short life compared to other devices (e.g. LED TVs) and one needs to determine what would be seen as reasonable in relation to a lamp's life. Now knowing the expected life of projector lamps, I am unsure whether I would purchase one myself as it would have a high operational cost (for lamp replacement) if the lamp needs to be replaced around 2000 hours.
For the average person that possibly watches TV for 18.25 hours per week (or up to another 13.5 hours per week if the internet/streaming is also projected as well), the lamp would need replacing every two years (or every 15 months if internet/streaming also occurs). Become quite expensive if the lamp is $171+ each time.