I feel you on having a team planned out and working around type coverage.
For the first gym if you're trying to avoid bug types and the rodent archetypes you're looking at usong some of these mons probably:
Budew event pokemon
Either the minccino or espurr event pokemon
Either the zebstrika or tyanmo
Also just because you expressed a bit of confusion at special encounters: Event pokemon in reborn are pokemon that have some sort of requirement other then walking in grass/cave/surfing. Some are as simple as interacting with a mon in the overworld and having a pokesnack, while others like the budew require certian weather or time conditions. Regardless these mons are evnt mons or special encounters because you can only get the mon from that event for either a long time until it shows up in the wild in a different area- or in rare cases its the only one you can find in the game.
With those four (starter included) at like a level 16-17ish and playing with switch style you could probably beat the first gym. You can always catch something to be a meat shield switch in, but really it depends on how fast you adapt to reborn style field effects and how they change battling along with your overall comfortability with pokemon.
Out of the ones mentioned the budew gives you roserade for a nice team mon which would tick off water, rock, ground, and fairy in addition to your fennikin ones.
After the first gym you don't get acsess to a whole lot more but once you get to the onxy ward you could pick up a luxray from the game corner as well as see what you get out of two eggs (likely to be a water pokemon) that you can find inbetween the first and second gym. Otherwise if you're not adverse to fighting types you can get a manky, pancham, or makuhita for a fifth or something like a buneary to get an early game loppuny that in the late game can be a mega
Between the second and third gym (beryll) you'll have acsess to the mystery egg sidequest which will give you 1 of 18 good mons- though some may not be to your liking or may conflict with a mon you already have. Either way its possible to slot something nice in from it.
Otherwise possible mons to get could be a shuppet from the graveyard to get ghost psychic coverage, or using a differnet electric type like mareep if you didnt pick one up ealier.
By the time you get to the lapis ward leader you'll have, if i recall correctly, acsess to all of reborn city which is where you can start to play more to your personal tastes and its harder for me to make recomendations. Looking at the potential team ive put together lacks you could grab a geodude/graveler or a sandlash.
Really though the best advice i can give is what the other poster brought up, is that trying mons you normally wouldn't can really suprise you and change your views. For me personally ive found that mons lile arbok, ratattata, grubbin, or many of the early game mons i personally regarded as not good put in some major work in my runs of reborn. Its made it so in the multiple runs of the game ive done i always try to do something new with a team.
Beyond that the only other advice I'd give is try to bring a team of six up to an even level as soon as you can, even if you replace some of them down the line or have to use duplicate types, before the first gym even. Also as the other poster mentioned some fights in the game may just striaght roll your team, and you might need to adapt it or bring ina new pokemon in order to make it manageable.
Still at the end of this i thought id put together a possible team that you could run and get going fairly early (before the lapis gym fight). If you yourself are trying to look at mons you can find id really try giving the spreadsheet or guide i linked another try, but theres also this forum for helping you find mons.
One caveat that i forgot to mention is that beyond just the normal morning, daytime, and nightime, many event pokemon are also have certian weather requirements or need to be fed a pokesnack in order to be battled or have them join. So if you find yourself unable to find a mon its worth checking for special requirments by use of this on the hunt forum or the other options you may find from googling. It can be worth it to get a weather changing mod if you are looking to either collect or catch all the different event pokemon you can find. If you're interested i can point you in the direction of that mod or you can find it through googling.
Grass, ice, bug, steel, fighting, poison
Water, rock, ground, fairy
Loppuny: (considered normal fighting due to moveset along with eventual dual typing from mega evolution)
Fire, electric, flying
Mystery egg pokemon:
Type pool left uncovered
I apologize if this was overly wordy or unhelpful, and if there any other specific mons you want to know where you can find in the game im happy to help out, and i hope you have fun with your runthrough of the game. Or if you need any clarification on something i wrote. Either way, cheers and best of luck.