If you have no interest in writing or reading the language, then you don't necessarily need to learn the Amharic alphabet. At the VERY least though, you're going to want a phonetic method for writing down the language that's consistent and accurate. By that, I mean that every single phoneme in the language has a different spelling.
If you were learning French, and your dictionary said that "su" was pronounced "soo" and "sous" was pronounced "soo" (when the two words actually use totally different vowels in French) then using that transcription method is going to set you up for major problems down the road.
Another potential problem with phonetic spellings in dictionaries is they tend to rely on English consonants and vowels that are similar, but not the same as Amharic consonants and vowels. Going back to the French example, "soo" doesn't actually line up with EITHER "sous" or "su", so every time you see the spelling "soo", you're going to have a tendency to pronounce that word with English rather than French vowels. It's the same idea with Amharic when using roman characters. If you choose to go that route, you'll likely have to fight that impulse the whole time. Will it set you back longer than it would if you learned how to write in Amharic? Hard to say, it is a complex alphabet. I usually find that you pick up a lot about how a language works from learning a new alphabet, and they're not as hard to learn as they appear.
When I'm not sure which way to go, I tend to lean towards the side of actually learning the original alphabet when possible.