Larger antenna may make it even worse because you will now receive even more interference.
I used to recommend inssider to see all the signals but they want money for it now. It is cheap but for a 1 time use it debatable of the value. If you can get the old one it is still free but it gets in the grey area of copyright violation to post links.
You can with effort see the surrounding SSID in the windows network selection screens and figure out which channels are used.
The best thing to try when you are in high interference is to try to reduce your requirements. To get 300 you must use 2 channel groups. If you force the router to only use 20mhz you will only use 1 channel and your odds of getting one with less conflict is better than getting 2 ie 40mhz. Especially if you are using 2.4g where there is only 60mhz of bandwidth total. I would set it to 20mhz and then try channel 1 or 11 first. Your max rate will drop to 150 but you may get more actually usable bandwidth.
The only solution you can use with antenna are to use directional antenna. They have narrow beams and tend to reject signal from outside the beam area better. Using direction antenna inside a house though is tricky because now you also have a narrow beam going though walls and floors and if something is in the wall that blocks it you have little ability to go around.
If it persists I would look at a non wireless solution. Obviously a ethernet cable is best but you could try powerline network if you have coax tv cable in both areas something like MoCA might be a option.