But since Max's nominal progenitor was/is/will be found at the Restaurant at the End of the Universe for some of the day, and at the Big Bang Burger Bar for the rest of it

From another left-hander and Arsenal fan (breedng will out):

The Restaurant at the End of the Universe is one of the most
extraordinary ventures in the entire history of catering. It has
been built on the fragmented remains of ... it will be built on
the fragmented ... that is to say it will have been built by this
time, and indeed has been -

One of the major problems encountered in time travel is not that
of accidentally becoming your own father or mother. There is no
problem involved in becoming your own father or mother that a
broadminded and well-adjusted family can't cope with. There is
also no problem about changing the course of history - the course
of history does not change because it all fits together like a
jigsaw. All the important changes have happened before the things
they were supposed to change and it all sorts itself out in the
end.

The major problem is quite simply one of grammar, and the main
work to consult in this matter is Dr Dan Streetmentioner's Time
Traveller's Handbook of 1001 Tense Formations. It will tell you
for instance how to describe something that was about to happen
to you in the past before you avoided it by time-jumping forward
two days in order to avoid it. The event will be described
differently according to whether you are talking about it from
the standpoint of your own natural time, from a time in the
further future, or a time in the further past and is further
complicated by the possibility of conducting conversations whilst
you are actually travelling from one time to another with the
intention of becoming your own father or mother.

Most readers get as far as the Future Semi-Conditionally Modified
Subinverted Plagal Past Subjunctive Intentional before giving up:
and in fact in later editions of the book all the pages beyond
this point have been left blank to save on printing costs.

The Hitch Hikers Guide to the Galaxy skips lightly over this
tangle of academic abstraction, pausing only to note that the
term "Future Perfect" has been abandoned since it was discovered
not to be.

QuiteEnoughDone