Assuming the value of the stocks held by the fund doesn't move, if I buy
a mutual fund 1 day before it pays a long term capital gain and sell
the fund 1 day after, do I get a long term capital gains on the
distribution and short term capital losses on the holding of the fund?

First, the taxation of funds would be described in the Prospectus.

Second, you have to say what kind of an account it is (i.e. tradition
brokerage, IRA, roth IRA, Keogh, etc.)

Third, the only periodic taxes you pay on a mutual fund is dividends
and interest  (although they are often reinvested, you do pay taxes on
them when earned, then going forward your basis would be adjusted
accordingly).

So, your question really doesn't make sense.  The amount of time a
fund holds a stock is really irrelevant, you are taxedon how long u
hold the fund, not how long the fund holds the underlying security.

Please note, foreign taxes are taken out when they are earne, but you
do not see it, it is reflected in the fund's performance.

Also, if this is in the variable insurance products that invest in
underlying funds, the rules may also be different and premium taxes
may also be applicable.

I see the situation differently.

The distribution of long term capital gains will be based on the
fund's holdings and activity and  - I believe -  reported as such to
the IRS and your stock broker, i.e., reported on your statement.  I
don't think the fund is in a position to differentiate that you have
have just bought into the fund.

Your stock broker will record and report your brief holding of the
fund, which may reflect an ex-dividend short term loss.

I doubt that the IRS will be able to or worry about connecting these two items.

If your statement shows a long term capital gains income, that would
be the logical way to report the income, and justification for doing
so.  If this is not correct, then it is certainly an excusable error.

As always, this is no professional or legal advice (see disclaimer below).