Searching the Balanced Corpus of Contemporary Written Japanese, I find the following results:
生きがい 606 results, 75% of total
生き甲斐 186 results, 23% of total
生甲斐 20 results, 2% of total
As you can see, in this corpus the most common way to write it is 生きがい. In fact, this is the form recommended by the NHK漢字表記辞典 (the kanji 斐 isn't included on the official jōyō kanji chart).
What about the alternatives? Well, writing 生き甲斐 appears to be somewhat less common, though I don't think there's anything actually wrong with it. Writing 生甲斐, though, appears to be significantly less common—this form has only 2% of the total search results!
Based on the above, I'd suggest sticking to 生きがい most of the time. But since you're doing calligraphy, I think you have to make an aesthetic choice for yourself. I'd just like to point out that hiragana can be very attractive in calligraphy.
The 連用形 (continuative form) of a verb can be used like a noun in Japanese, and it's not uncommon for this form to be lexicalized as a noun or as part of a larger compound noun. Since nouns don't inflect, these derived forms don't really need okurigana most of the time, and there's a tendency to get rid of them. For example, 取引 or 話 are generally written without okurigana, even though they're clearly derived from the continuative forms 取り＋引き and 話し.
But the orthography isn't really consistent this way. And some words, like 後ろ, use okurigana even though they don't inflect, in order to prevent misreadings. Tsuyoshi Ito left a comment suggesting that something similar might be the case here as well, that 生（いき） might be avoided because of 生（なま）. I don't know myself whether or not this is the case, but we can see from the corpus results that even though it's a lexical compound and doesn't inflect, people usually don't write 生き甲斐 without okurigana.
Based on the corpus data and Tsuyoshi Ito's comment, I think I would avoid writing 生甲斐 under most circumstances.