One or the Other, Not Both
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You can either make the drive in 5½ days and maybe make one or two relatively quick stops OR you can take a 'southern' route and maybe avoid winter weather, but you can not take the southern route and have time to see anything, or have time to deal with any inclement weather you do meet. The fact is that staying south of the main portion of the Rockies and going all the way to Los Angeles before turning north results in a trip length of over 3,100 miles - close to 6 full days of driving. And that comes with no guarantee of good weather. Both I-40 and I-10 have already seen snow and sleet this season and could see more. If I were trying to set up this trip, I'd plan on taking the direct 'diagonal' across the country, I-75/I-24/I-64/I-70/I-29/I-80/I-84, which is only 2,600 miles and use the 'extra' time that buys me as a hedge against bad weather, or if the weather co-operated, as sightseeing time.
AZBuck