Data analysis of æ (tense) and æ: (lax) (Lexical Phonology)

I am guessing this is a UK dialect, since I disagree with a number of the factual claims. There are major gaps in the data, which should mean you are free to predict either outcome in those contexts. Overall, your analysis looks "rightish".

It is fairly well-known that there is a general vowel shortening triggered by voiceless coda stops, and that the shortened vowels are somewhat higher than the "regular" vowels. This applies to all vowels, and results in 4-way surface differences like beat/bead/bit/bid. This is generally "allophonic" and non-neutralizing, so it isn't necessarily phonological as opposed to being phonetic implementation -- the two words "adder" would be crucial in that respect. Your analysis doesn't mention the syllable. Part of what makes this problem a bit odd is the difference between VCV sequences within words versus between words (VC#V). But we also know that there can be (dialectally) special syllabification in the context VC#V especially with unstressed following words. The question is: if you frame the analysis in terms of syllable structure and not CV sequences and if affixation or phrasal concatenation can result in different syllabifications, how does your analysis change?

More than once, I have heard the argument that equates frequency with proof of underlying basicness, and as many times I have argued that that argument is invalid, unless you stipulate it as an axiom of analysis (LP does not). The first question is why you think the underlying representation and the surface representation are not the same: what forces you to reduce the surface representations (SR) to more abstract underlying representations (UR)? Do you think there is some mandate to reduce the number of segments in underlying forms?

The idea that distribution is sensitive to lexical or morphological is factors is a conceptual error. The thing (rule) that causes a distribution may be sensitive to such factors, but the distribution is itself completely insensitive. In classical Lexical Phonology, the rules are not sensitive to lexical or morphological factors. The rules simply apply when the required structures are present. If within a single level you still had a rule which distinguishes certain morphemes which trigger / undergo the rule from ones that don't, then you could say that the rules are sensitive to morophological or lexical factors. But you don't have that. Instead, you have a rule that applies in a specific level, but doesn't care about the M/L factors. Level ordering generates behavior that sometimes resembles morpheme-specific conditioning.

The question about contrastiveness is meaningless without a definition of contrast. If "contrast" means "both are present in the dictionary", and you have an axiom of phoneme-elimination from the dictionary, then the two variants of æ would not be contrastive (not both found in the dictionary). Otherwise, they are contrastive. The argument that "the words are at different levels" is wrong. The output is a level; both words exist at that level; they are minimal pairs at that level. It does not matter that they are "brought about" at different levels. As always, invoking "contrast" without a definition of what "contrast" is is a mistake.