So the rules on variant classes are fairly minimal. They were introduced in Unearthed Arcana, a collection, basically, of ideas a DM could use to add variety to their game. Some are more and some are less fleshed out, but none of them really purports to be complete, done, polished like content in other books.
Then other books used the same concept without doing any of the fleshing out that Unearthed Arcana skipped.
Anyway, here is what we do have on this subject:
Multiclassing between variants of the same class is a tricky subject. In cases where a single class offers a variety of paths (such as the totem barbarian or the monk fighting styles), the easiest solution is simply to bar multiclassing between different versions of the same class (just as a character can't multiclass between different versions of specialist wizards). For variants that are wholly separate from the character class—such as the bardic sage or the urban ranger—multiclassing, even into multiple variants of the same class, is probably okay. Identical class features should stack if gained from multiple versions of the same class (except for spellcasting, which is always separate).
Note this is more musing on what you (as DM) might do, rather than a hard and fast rule. But if we assume your DM implements this suggestion, we can probably extrapolate from there with some confidence. To wit, evangelist appears to be “wholly separate” from cloistered cleric, much more so than bard and bardic sage are from one another, so Unearthed Arcana would seem to support one character multiclassing both of them. You will still have to check with your DM, though.
Would this result in losing a caster level?
Yes, this one Unearthed Arcana is pretty clear and emphatic about:
Under no circumstances does spellcasting ability from multiple classes (even variants of the same class) stack. A character with levels of bard and levels of bardic sage has two separate caster levels and two separate sets of spells per day, even though the classes are very similar.
Your evangelist and cloistered cleric spellcasting are separate, so if you are a cloistered cleric 1st/evangelist 1st, you have caster level 1st in each, with separate spells per day for each. Making the cloistered cleric also a spontaneous divine caster doesn’t change anything.