It seems to me that your client, who is in the travel business, should be TELLING you which booking system they want to use, rather than ASKING you, bearing in mind you don't know anything about these systems. (And neither do I, by the way).
Surely the choice of a travel booking system is a fundamental business and commercial decision for the travel business owners, not something to be palmed off to the website designer? You will quickly be blamed for all the inevitable perceived shortcomings of the system you recommend based on such minimal information, that's for certain!
Let your client do their own research and come up with a list of possibilities. They don't sound very serious, if this is how they choose their core business systems. I strongly suggest staying well within the limits of your own expertise. Personally, I wouldn't touch this job with a bargepole. It's got 'imminent disaster' written all over it :badpc: