How do you know that 350' is not going to work? What (minimum) bandwidth does the cam need? What are the cam's power specs?
100M length is a standard, to keep the world spinning and get things done, but that doesn't necessarily mean that you can't get more than 100M to work.
I suspect there is a good chance of it working if one of the following is true, but it depends also on how much "a little more" distance actually is, and whether you can accept a little packet loss. #2 and 3 below could require a little hands-on with wire stripping/soldering/connectors/etc
1) High quality all copper cable, the lower the gauge the better like 22AWG or lower.
2) Run a separate POE power feed cable, 22AWG or lower. You can test power loss or calculate it based on wire gauge and length, assuming true copper conductors not copper clad aluminum. It need not be ethernet cable. With a mock-up and the cam connected, you can measure power loss and test functionality in operation before the installation.
3) Instead of the lower gauge of #1 above, run either a separate ethernet cable for power delivery, or if running at 100Mb instead of Gb which will increase the usable cable length, -either way- the extra conductors not used with 100Mb using one cable, or conductor wires in a 2nd ethernet cable, can be used for paralleled power conductors, two or more wires in parallel for POE +, and two or more for POE -. POE distance is really about the resulting volt drop over X length of Y gauge wire, and what voltage range the device with POE, or a separate POE module can use and still output enough power for the cam if it has a separate DC input socket.
4) Use something like this Gigabit POE extender repeater instead:
#4 is better than having to use an enclosure for a bulkier and potentially more power hungry multi-port POE switch, though in harsh weather, an enclosure still wouldn't hurt. There might be better brands/models of extenders, I haven't looked into it, but the same brand also has more related options, possibly also a similar extender repeater that isn't outdoor rated for lower cost. One on the 2nd link is currently sold out but may be available elsewhere:
Buy PoE+ and Ultra-PoE injectors and extenders by Intellinet. Free shipping on orders over $49 within the US.
intellinetsolutions.com
Gigabit High-Power PoE+ Extender Repeater, IEEE 802.3at/af Power over Ethernet (PoE+/PoE), metal - Retail Box
intellinetsolutions.com
I was also going to suggest some alternatives for power at the other end like a solar panel, charge controller and battery (ideally LiFePO4 for high cycle life), testing needed to determine solar input vs battery size vs cam requirements vs climate (snow blocking panel in winter) but you are
*so close* to this working without that, that I'd consider it a last resort. I don't like to depend on the variability of daily sunlight if it can be avoided.
However, this brings up an interesting idea. How much is this cam going to be active during a day, particularly in the dead of night? Is it always on or motion activated or ??? A weak POE running 24/7 to recharge a battery that powers the cam, if the cam isn't at full power consumption 24/7, could also work. Daily POE delivery to the battery then just needs to exceed daily consumption by the cam, plus efficiency losses charging the battery and then buck or boost regulating it to the DC input the cam needs.
Obviously these wouldn't be off-the-shelf computer networking components, but still a possibility. Granted that doesn't extend the data distance capacity, but it could be a matter of handling data vs POE separately, hooking up the cabling then seeing where there are problems if there are, unless time = money then just get an extender module as mentioned in #4 above.