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Here not an answer but a reviewed version of your code because I am getting a different result than the one you posted. Please show us where I am wrong or edit your question:

from rdkit import Chem
from rdkit.Chem import Draw

def check_bredts_rule(molecule):
    ring_info = molecule.GetRingInfo()
    atom_rings = ring_info.AtomRings()

    atom_ring_counts = {atom.GetIdx(): 0 for atom in molecule.GetAtoms()}
    for ring in atom_rings:
        for atom_idx in ring:
            atom_ring_counts[atom_idx] += 1

    bridgehead_atoms = [atom_idx for atom_idx, count in atom_ring_counts.items() if count > 1]

    
    for atom_idx in bridgehead_atoms:
        atom = molecule.GetAtomWithIdx(atom_idx)
        is_part_of_alkene = any(
            bond.GetBondType() == Chem.BondType.DOUBLE
            and bond.GetBeginAtomIdx() == atom_idx
            and bond.GetEndAtom().GetSymbol() == "C"
            for bond in atom.GetBonds()
        )
        if not is_part_of_alkene:
            continue  

        ring_sizes = [len(ring) for ring in atom_rings if atom_idx in ring]

        if all(size < 8 for size in ring_sizes):
            return 'violate'

    return 'no violate'


mol1 = Chem.MolFromSmiles('C1C[C@H]2C[C@@H]1C=C2') #Bicyclo[2.2.1]hept-2-ene NO VIOLATION

mol2 = Chem.MolFromSmiles('C1CC2C1CC=C2') #Bicyclo[3.2.0]hept-2-ene NO VIOLATION

mol3 = Chem.MolFromSmiles('CC1=C2CC[C@@]2([C@H]3CC(C[C@H]3C1)(C)C)C') #Delta(6)-protoilludene VIOLATE

mol4 = Chem.MolFromSmiles('CC1=C2C[C@](C[C@H]2CC3=COC(=C13)C(=O)O)(C)CO') #Tsugicoline A VIOLATE

mol5 = Chem.MolFromSmiles('C[C@@H]1CC[C@@H]2CC3=C(CC[C@]13C2(C)C)C') #Cyperene VIOLATE

mol6 = Chem.MolFromSmiles('O=C1C2CCCCC1CC2') #Bicyclo(4.2.1)nonan-9-one No violation

mol7 = Chem.MolFromSmiles('O=C1CCC2CC1CCC2=O') # Bicyclo(3.3.1)nonane-2,6-dione no violation

mol8 = Chem.MolFromSmiles('O=C1C2CC3CC1CC(C2)C3=O') #2,6-Adamantandione no violation

mol9 = Chem.MolFromSmiles('C1CC2CCC1=C2') #violate


mols = [mol1, mol2, mol3, mol4, mol5 , mol6, mol7, mol8 , mol9]

for mol in mols:
    
    print('\n\n', Chem.MolToSmiles(mol) ,'--> ', check_bredts_rule(mol))
    
    

img = Draw.MolsToGridImage(mols, molsPerRow=3, subImgSize=(200, 200), legends=[check_bredts_rule(mol) for mol in mols], 
                           highlightAtomLists=None, highlightBondLists=None, useSVG=False, returnPNG=False)


img.show()

output:

C1=C[C@H]2CC[C@@H]1C2 -->  no violate


 C1=CC2CCC2C1 -->  no violate


 CC1=C2CC[C@]2(C)[C@H]2CC(C)(C)C[C@H]2C1 -->  no violate


 CC1=C2C[C@](C)(CO)C[C@H]2Cc2coc(C(=O)O)c21 -->  no violate


 CC1=C2C[C@H]3CC[C@@H](C)[C@]2(CC1)C3(C)C -->  violate


 O=C1C2CCCCC1CC2 -->  no violate


 O=C1CCC2CC1CCC2=O -->  no violate


 O=C1C2CC3CC1CC(C2)C3=O -->  no violate


 C1=C2CCC1CC2 -->  violate

with image:

I am not a chemist, but according to this Anti-Bredt Olefins: Never Say Never! your mol9 should give a NO VIOLATE output

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