No offense, but I think your playstyle is fundamentally incompatible with how the mod is structured, so the easiest solutions would be to change your playstyle to adress the issues you are facing. "Affects game progression" doesn't mean you aree being gated off from research. You will have more than enough opportunities to research items/captives/etc even if you don't research them immediately. You can postpone most non-progression related research, but you can not indefinitely postpone research that is critical for (early) progression.
If that is not an option I see multiple approaches to this, unfortunately all of them would involve editing game or save files. While playing on beginner would make battlescape missions easier I am not sure it would make tackling bases any more fun, nor would it prevent larger craft from being intercepted. Still, playing on beginner would be preferable becasue you are likely going to be outmatched anyway.
If you simply want to postpone the invasion (and cut down on cult spawns and other missions) you can reset the number of elapsed months in your savegame via "monthsPassed: " every new month and essentially stretch out the game. Note that this likely isn't going to work with your ongoing game, as it will not despawn cult bases.
If the bases are what is giving you trouble you can delete under "alienBases:" in your save game. Note that this is likely only a stopgap measure and the bases will keep spawning unless you terminate the cults. It will also not net you any positive score, it will only cut down on the points the aliens score.
If countries being unhappy is your main concern you can also modify points for xcom and alien activity in your save under "activityXcom:" and "activityAlien:" on a per-country basis.
If you are mainly missing funds you can edit them too under "funds:".
Note that for all of these the most recent value (the one for the ongoing month) is listed at the bottom.
If you simply want to reset the negative score warning set "warned:" to "false".
If research speed / Number of topics are the problem you could decrease the points required for certain topics in research_XCOMFILES.rul or you could increase the number of lab space facilites provide in facilities_XCOMFILES.rul (look for "labs:"). Overall I don't think there is a convenient or straight-forward to achieve what you are asking for, simply because the mod isn't designed for "laid back" kind of play, at least not early on when you're missing most of the transports, armors, equipment etc.
It gets better as you progress, but shutting down the cults and getting promo II/III as fast as possible is pretty much mandatory, otherwise you will get swamped in cult bases, as you experienced.