Modern Standard Arabic (MSA) won't help you directly in speaking, but it will help you in every other aspect of the language.
Your children will grow up knowing how to deal with Arabic. Now that means they will speak the dialect they hear, they will read books and newspapers in MSA, and they will watch the news in MSA.
Generally I would recommend starting with the closest dialect to your husband's, but in your specific case I think MSA will be useful in the beginning. You don't need a high level in MSA; just the basic rules for B1 level would be sufficient. This way, you will get much faster into the dialects, since the dialect that you are dealing with is a mixture of two main dialects; Levantine Arabic & Iraqi Arabic.
Grammar is the same for all dialects and MSA; this is one way MSA can save you time when learning the dialect, since you will be focusing only on vocab. Levantine Arabic has different accents; Syrians can know this is Lebanese Arabic, and Lebanese can know this is Syrian Arabic. For other parts of the Arab world they cannot distinguish between them. That's a bonus that your husband was raised in both Syria and Lebanon, because this means you will not care about accurate Syrian accent or Lebanese accent; any Levantine accent will be perfect when talking with your husband and your children.
I recommend starting with MSA to know the basic rules, shift to Levantine Arabic since it is easier and richer in materials (almost all foregin TV series and movies that are dubbed in Arabic are using Levantine Arabic), then you can go get some knowledge about Iraqi Arabic as it is not as widespread as Levanitne.