I have the tools to install MCX, SMA and BNC connectors on this type of small RG174 coax used with these antennas. But JonJay nailed it. The small components are hard to work with and 9 out of 10 of them require soldering with a super fine tipped soldering iron in a small cavity. You have better have a magnifier glass and zero hand shake (no coffee for you that day). Since these are active antennas a bad connector installation could short and fry something expensive. If you had a BNC end it would be easier as there are double crimp connector options not requiring soldering. But yours appears to be MCX.
Sam hit on another very good point. The designers of the components use an assumption of a minimum amount of impedence loss in a typical installation. To reduce the db loss with a shorter run on a lead can cook components. For example, when installing UAT out ADS-B units Freeflight requires a minimum loss so I have gone so far as to use RG 58 and even RG174 coax lead to the ADS-B transponder type antenna in order to meet the minimum loss rather than coiling up a longer run of higher quality RG400 coax required to achieve the same loss. Interestingly the Garmin GDL-82 doesn't specify a minimum loss value and so I keep those coax runs short. As for your replacement item, I can't answer as to whether the GPS antenna designers need the full 10 feet of coax to meet any loss requirements.
In the end I always keep the as delivered length and neatly coil up the excess cable in a round loop with zip ties behind the panel and have never had any signal or performance issues.
Jim