Confusion in the order of precedence: alkyl side-chain vs. dimethyl at different positions
Your starting point is correct. According to the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book), the name 5-(sec-butyl)-2,7-dimethylnonane is no longer recommended. (The prefix ‘sec-butyl’ was still contained in the 1993 recommendations.)

The names that are proposed in the question, 2,7-dimethyl-5-(1-methylpropyl)nonane and 5-(1-methylpropyl)-2,7-dimethylnonane, are unambiguous and describe the same structure. There may be many different ways to draw the structure of such branched alkanes. If you enter these names into the Open Parser for Systematic IUPAC nomenclature (OPSIN), which is mentioned in the question, the results are as follows:


2,7-dimethyl-5-(1-methylpropyl)nonane


5-(1-methylpropyl)-2,7-dimethylnonane

The structures may look different at first glance, but they actually describe the same compound. In this respect, there is no difference between 2,7-dimethyl-5-(1-methylpropyl)nonane and 5-(1-methylpropyl)-2,7-dimethylnonane. However, the name 5-(1-methylpropyl)-2,7-dimethylnonane is not in accordance with IUPAC nomenclature since ‘methyl’ is earlier alphabetically than ‘methylpropyl’. Therefore, the name 2,7-dimethyl-5-(1-methylpropyl)nonane is a correct IUPAC name.

Anyway, the preferred IUPAC name (PIN) according to current recommendations is 5-(butan-2-yl)-2,7-dimethylnonane.