Even with the restriction you propose, there are potential balance problems. Your DM will have to agree on any list of approved additions to Beasts, as well as a way to scale down the at will powers (super cantrips?) a given Monstrosity or Aberration has. Before I get into analysis:
Second, I'm giving up the druid spellcasting abilities to make up for
the magical abilities that I'll be given access to.
A step in the right direction. You still have spells, at will, at levels of varying power.
Some at will abilities are very powerful. I'll offer one example with your Int restriction included to illustrate. I suggest you do a similar analysis for each candidate creature you identify and propose it to the DM, with any suggestions for power mitigations for at will abilities.
Actions Bite. Melee Weapon Attack: +5 to hit, reach 5 ft., one target. Hit: 10 (2d6 + 3) piercing damage plus 7 (2d6) poison damage.
Other beasts do decent damage and poison. (CR 3 Giant Scorpion, for example).
STR 16 (+3), DEX 8 (−1), CON 15 (+2), INT 2 (−4), WIS 8 (−1), CHA 7 (−2)
Your INT, CHA, and WIS carry over, so you don't pay the penalty on saves or ability checks that an ordinary basilisk has. (OK, a basilisk is hardly ordinary ... ) Carry over is a feature of Wild Shape.
Senses darkvision 60 ft., passive Perception 9
If human, free darkvision at no spell cost. Some beasts see-in-the dark, no problem.
Languages — (none)
(Basilisks were never much for small talk). :-)
The at will ability that needs work.
If a creature starts its turn within 30 feet of the basilisk and the
two of them can see each other, the basilisk can force the creature to
make a DC 12 Constitution saving throw if the basilisk isn’t
On a failed save, the creature magically begins to turn to stone and
This is an at will ability (like a cantrip) in the form of a spell effect that is at least 3rd level, per the cockatrice in the other question. At least as powerful as Hold Person with no need to concentrate, except that the DC is lower than a spell that you'd cast with your WIS bonus as a Druid.
Note that this isn't an Action: it's like an aura. (Or maybe a bonus action?). It's like having a wand or staff of this ability with dozens of charges. Magical items are scarce in 5e, and here you can turn into this thing and get a wand's worth of stuff ... and don't need an Action to make the magic work! Power a-plenty, could need scaling back.
Just standing around, for the duration of your transformation, you can expend an equivalent of most of your spell levels in power, maybe more, with this feature alone. This with a creature you can become as a 9th level CoTM Druid.
It must repeat the saving throw at the end of its next turn. On a
success, the effect ends. On a failure, the creature is petrified
until freed by the greater restoration spell or other magic.
While it takes 2 failed saves to turn to stone, a statue is out of the fight for good. (More on that later). If one of your party casts Bane and some saves are at Disadvantage you could make multiple statues with some haste. Fight's over. It takes a fifth level restoration spell per creature to undo this change in status. (On the other hand, your garden decoration needs would be solved, so that's a plus).
In terms of Encounter Balance: you spend no spell slots and require of your enemy the cost of a fifth level spell to get back into the fight.
A creature that isn’t surprised can avert its eyes to avoid the saving
throw at the start of its turn. If it does so, it can't see the basilisk
until the start of its next turn, when it can avert its eyes again.
If it looks at the basilisk in the meantime, it must immediately make
My DM has us attack at disadvantage if we avert our eyes during combat to avoid things like an Umber Hulk's gaze/Confusion. (Related to RAW on targets you can't see ...) Against this Basalisk-Druid, foes at melee and short-spell-or-missile range will have disadvantage just because you are there. No need to cast any sort of spell (like the Cleric's Bane) to inflict it upon them. No need to poison them and have them save versus poisoned condition. (They may already be doing that against your bite!)
Encounter Balance problem number 2: you get free 1st-or-2d second level spell effect slots for as long as the fight lasts but spend no spells.
How many CR3 beasts have that edge in a fight?
How do you scale down the flesh-to-stone super-cantrip that operates in an area of effect (3d / 4th level power)? Let's look at some cantrip level powers, Wizard:
Acid Splash, Fire Bolt, Mage Hand, Minor Illusion, Poison Spray, Ray of Frost, Shocking Grasp
If you can nerf the gaze into that power range, or a lower level spell, this might fit a bit better in the case for a Monstrosity.
If the basilisk sees its reflection within 30 feet of it in bright
light, it mistakes itself for a rival and targets itself with its
Huzzah, a weakness for the DM to exploit! You'll run into an enemy fighter with a shield that is polished to a mirror finish! :-) (Perseus' Shield from Greek mythology, versus Medusa).
A basilisk is a mult-ilegged, reptilian horror whose deadly gaze
transforms victims into porous stone. With it strong jaws, the
creature consumes this stone, which returns to organic form in its
Second order effect and a nasty bonus power. You can turn your enemies to stone, eat them, and then head to the latrine to dispose of them. This isn't quite as powerful as Disintegrate spell, but it's up there. How many fourth and fifth level spells per day does a Druid get at 9th level? At 11th level? At 15th level?
Scale down the gaze to a bonus action between "Ray of Frost" to a mini "Cone of Frost" power levels, an effect that slows down your foe, maybe multiple within (___) range. You already get free disadvantage with your gaze. Its' still a free spell, and a spell you didn't have to choose or prepare, in terms of equivalence. (Oh, and you aren't really playing a basilisk anymore). Between you and your DM to work out the particulars.
A similar sort of analysis should be done on a per-creature basis to get the balance into the ball park. Work with your DM.