Should I use Me or Myself? Help

Normally, it should be the reflexive pronoun, myself. A possible exception is if you wish to create some sort of a special emphasis; see the quote from CGEL, below.

A simplified sentence

Consider the simplified sentence

[1] I make him and myself/*me happy.

Here the noun phrase (NP) him and myself is the direct object of the verb make, while happy is a predicative complement (PC). (For what it's worth, CGEL characterizes this construction as being 'complex-transitive with obligatory resultative PCs' (p. 265).)

The case of a non-coordinated object

Let us consider first an even simpler case:

[2] I make myself/*me happy.

Here the pronoun myself is the direct object of the verb make. The antecedent of this pronoun is the subject of the sentence, I. In such a situation, when the direct object is a pronoun whose antecedent is the subject of the sentence, use of a reflexive pronoun is normally mandatory (CGEL, p. 1487).

Override

Now, there is such a thing as override, where we use a non-reflexive version where a reflexive one is normally required. This is pretty rare; the converse case, where we use a reflexive instead of normally required non-reflexive, is much more common. Here is what CGEL says about the non-reflexive override (p. 1485):

Override can also work in the opposite direction, with a non-reflexive form appearing instead of the normal reflexive:

[6] Why don't you buy something for Y͟O͟U͟ for a change, instead of spending all your money on your kids?

The normal form here would be reflexive yourself: non-reflexive you is used to emphasize the contrast between you and your kids, though reflexive forms can also be used contrastively. It was to allow for such overrides that we defined mandatory reflexives as those which cannot normally be replaced by the non-reflexive counterpart: we take the reflexive in You bought it for yourself to be mandatory, clearly different in status from the optional reflexive of [4ii] [Ann tied a rope around herself/her.]. Override non-reflexives like [6] are largely restricted to informal style, and much less common than override reflexives: we need not consider them further.

So it seems safe to assume that the only way we could use me in [2] was if we wanted to create some special effect, like an emphasis.

Coordinated objects

Now, [1] is a bit more complicated than [2] because the object is a coordination, him and myself. And it is known that in English grammar, sometimes what is true for non-coordinated things is not true for coordinated ones. For example, while you are eligible may be contracted to you're eligible, this cannot be done if you is coordinated: we must say Kim and you are eligible, and cannot say *Kim and you're eligible. Similarly, while we must use the accusative me in They invited me to lunch, if the object is coordinated this rule gets relaxed a bit. Namely, a significant proportion of speakers of Standard English would say They invited my partner and I to lunch, and this is passed unnoticed by the rest of speakers all the time, even in broadcast speech (CGEL, p. 9).

Having said that, cases like the above (where the form of a word entering a coordination may be different from the form it takes when not in coordination) are not that common. So the burden of proof is on those who would claim that, in common usage, the form of a word does (or can) change depending on whether it enters a coordination or not. Edwin Ashworth, in the comments above, gave one example of how that can happen: although we normally must use me in Hand in your essays to me, when the object of the preposition is coordinated, it becomes quite acceptable to use the reflexive form: Hand in your essays to Jean or myself. Note that this is the more common type of override, where the reflexive ovverides the non-reflexive; in our case it would be vice-versa. Nevertheless, we should check what kinds of patterns of usage exist in cases most relevat to us. Which brings us to...

Evidence from published literature

If we search google books, we get hits like

All I'm saying is that I will date and make other people and m͟y͟s͟e͟l͟f͟ happy.
I would again make him and m͟y͟s͟e͟l͟f͟ proud.
I wanted to make the kitchen—and m͟y͟s͟e͟l͟f͟—proud of the meals I would create here.
I did my parents and m͟y͟s͟e͟l͟f͟ proud.

I haven't been able to find clean examples where myself is replaced by me (where the antecedent is the subject of the sentence, I). I suspect they are there, perhaps with some adjective other than happy, proud, and several others I tried, but I haven't been able to find them.

In conclusion, almost certainly the correct choice in your case is myself.