I found this website for some ammo if you need it. For plans with less than 100 participants the limit is 7 days after deduction from your paycheck, though an extension is possible with some paperwork. A DOL audit would probably land your employer in some trouble if they are regularly taking 4 weeks to deposit your funds in the 401k account.
Part of your first link has this statement that I suspect you are missing:
However, Section 13(a)(1) of the FLSA provides an exemption from both
minimum wage and overtime pay for employees employed as bona fide
executive, administrative, professional and outside sales employees.
Note that executive is in that list. As for the additional note:
The 401(k) plan must have a definition of "eligible employees". All eligible employees must be given the opportunity to participate in the plan.
Talk to the 401(k) contact within your company, or the 3rd party Plan Administrator. Ask to see the plan documents. Learn how it defines "eligible employees" and find out if part-time employees are eligible or not, ...
Even under the executive exemption, see Exemption for Executive Employees Under the Fair Labor Standards Act (FLSA) Section 13(a)(1) as defined by Regulations, 29 CFR Part 541, it seems that a minimum compensation is required.
To qualify for the executive employee exemption, all of the following tests must be met: The employee must be compensated on a ...
Based on the info provided in the comments in the question the following timeline seems to apply:
Quit old job in July.
Lose health coverage from employer on July 30th
New employee coverage starts September 1st.
Need to decide about COBRA in September.
Because COBRA is retroactive, it takes weeks to get the forms to you and they give you time to decide, ...
The amount on the LCA is the minimum they can legally pay you, it's no problem to pay more, you have income tax obligations on the full amount of your pay, of course.
Here are a couple snippets from CFR › Title 20 › Chapter V › Part 655 › Subpart H › Section 655.731 that indicate the LCA amount is a minimum:
(3) Once the prevailing wage rate is ...
If an employee created IP with his own resources, not in working hours and outside the workplace, then there is no reason for the employer to claim the IP. But in some cases, some companies want to own IP created outside work if the employee develops IP from the knowledge he has acquired while working, or if the IP developed can be used by competitors, to ...