You are correct, inheritance is not a supported way just like there is no switch from not required -> required. We lack a way to define a TypedDict partially.

Currently you have to define your TypedDict twice, once for each totality.
Functional vs. class definition does not make much of a difference. From you question I assume that you want to reuse a dict variable, e.g. Def = {"foo": int} and TypedDict("DefTotal", Def, total=True), while this would allow you to easily keep both dict types in sync it is not required for type-checkers to accept it (per PEP 589):

a variable that refers to a dictionary object does not need to be supported

Neither mypy not pyright allow dict variables and expect a literal dict for the functional syntax.

TL;DR: There is currently no supported way to reuse the key: value definitions and change between total <-> not total. You have two literally define both cases.

For completion, there is a possibility of a generic TypedDict[T: (Never, Missing)] where you make all keys required but with a Missing default value (some custom Sentinel), i.e keys are foo: int | T. Subscribing such a TypedDict with Never (complete: YourTD[Never]) gives all keys the correct type, i.e. complete["foo"] # -> int.
While it works it is less intuitive and you need to juggle around with subscriptions and/or some aliases.
Using Never as the default value for the type variable might also be a good idea but more risky than Missing which is the safer default to be recommended. If the sentinel type is annoying you can also substitute it with Any for the Any-Trick.

Code sample in pyright playground

from typing import TypedDict, Never, TypeAlias
from typing_extensions import Sentinel
MISSING = Sentinel("MISSING")
# Aliases for clarity
AllKeysPresent: TypeAlias = Never
MaybeMissingValues: TypeAlias = MISSING
# Never as constraint can be omitted, likewise total=True
class LockeableDict[T: (MaybeMissingValues, AllKeysPresent)](TypedDict, total=True):
 # Define all keys with T
 foo: int | T
PartialDict = LockeableDict[MaybeMissingValues]
FinalDict = LockeableDict[AllKeysPresent]
A: PartialDict = {"foo": MISSING} # all keys required
# make sure no missing keys 
# ...
B: FinalDict = A # <-- need to cast
reveal_type(A["foo"]) # int | MISSING
reveal_type(B["foo"]) # int
A["bar"] = 2 # error