Orthographic variation is very common in Luxembourgish texts due to the absence of a fully-fledged standard variety. Additionally, developing NLP tools for Luxembourgish is a difficult task given the lack of annotated and parallel data, which is exacerbated by ongoing standardization. In this paper, we propose the first sequence-to-sequence normalization models using the ByT5 and mT5 architectures with training data obtained from word-level real-life variation data. We perform a fine-grained, linguistically-motivated evaluation to test byte-based, word-based and pipeline-based models for their strengths and weaknesses in text normalization. We show that our sequence model using real-life variation data is an effective approach for tailor-made normalization in Luxembourgish.