Yes, ne rege absente sacra neglegerentur is a purpose clause.
iturosque ipsos ad bella – -que joins this clause with in civitate bellicosa plures Romuli quam Numae similes reges fore, and both clauses are complements of putabat.
ituros is, as you noted, a future participle–it’s part of a “periphrastic” future infinitive with esse omitted (as it often is in real Latin). The subject of this future infinitive must be [reges] ipsos.
So we have "But because he believed that . . . there would be more kings similar to Romulus than to Numa, and they would go to war themselves, . . . "
ne rege absente sacra neglegerentur, as you noted, is a purpose clause, and you got the purpose clause more or less right the first time, except that neglegerentur is passive: lest . . . the sacred rites should be neglected"
rege absente – you translated this “through the absence of a king”. I’m not sure you recognized that this is an ablative absolute. I think a better translation would be “a/the king being absent/away,” or “while a/the king was absent/away” or just “in the king’s absence”.
in civitate bellicosa – you translated this “amongst the warlike citizens”, but civitas is singular, not “citizens” but rather “state,” “body of the citizens” or “community”. Here I would translate it “in a warlike community”.
assiduum – take a look at the meanings given by Lewis & Short. I would translate here “permanent”.
http://perseus.uchicago.edu/cgi-bin/philologic/getobject.pl?c.0:4065.lewisandshort
So, we have:
Sed quia in civitate bellicosa plures Romuli quam Numae similes reges fore putabat iturosque ipsos ad bella, ne rege absente sacra neglegerentur, Iovi assiduum sacerdotem creavit qui ‘flamen Dialis’ appellatus est.
“But because he thought that in a warlike community there would be more kings similar to Romulus than to Numa and they would go [off] to war themselves, lest the sacred rites be neglected while the king was away, he created a permanent priest for Jupiter who was called the flamen dialis.”
At this point, I would suggest you shouldn’t try to recast your translations into more idiomatic English until you have translated in a way that clearly reflects the syntax of the original. The object isn’t to translate idiomatically–it’s to understand how the original Latin is structured so that you master the Latin constructions. If you’ll allow me to offer some advice, I feel that too often you assign English meanings to the words and then try to make sense out of the whole sentence without fully working out the syntax. You can’t understand the meaning of a sentence without understanding the syntax. Not only do you need to pay close attention to the inflectional endings of nouns, pronouns, adjectives and verbs, you have to think about how they fit together. What noun does this adjective modify? What is the function of that noun phrase in the sentence? What is the mood, tense and voice of this verb? Why is it subjunctive? What is its subject and what is its object? If you follow this practice very rigorously, after a while it will come naturally and you won’t have to think about it.