How do i balance MI requirements with the user experience?

In this case UX should always trump information gathering. There are plenty of indirect ways of gathering information about your users without antagonizing them. Interaction with your users should follow social rules. Just like a website should not ask more information than a person would, a website should not pretend to offer options it doesn't have.

Your users will see your website as a jerk. No amount of information gathered about them will balance out that damage.

As an alternative, I'd recommend the following:

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This approach gives your users only the options that work as expected. The unavailable option is clearly marked differently. The user can see that opening the textblock will not destroy the progress in the form, so they are likely to open the box if they want the alternative option.

I've also added a text box to let users give you feedback. If you ask users for feedback directly and honestly, at the moment they're faced with the problem you're interested in, they will be more than happy to give it to you.

If you don't like the feedback option, you can also use an AJAX call to check how often people open the text block. It will give you an indication of the interest in other options than the available four, and the users don't get unexpected behavior.

You can try to smooth user experience by doing following:

1. Replace a lightbox with separate page containing detailed information about this option, call centre phone number and "back to options" navigation button. It will let users think that they are moving forward to get the desired option.
2. Use positive wording ("can" instead of "cannot", "available by call" instead of "not available online" etc)
3. If possible, provide alternative means for getting assistance from customer care (request a call, contact by email etc). This may cost less than implementing the option online and will be more attractive to your management as temporary solution.

Yes, that's annoying all right, no way around it that I can think of. You could try and alleviate any annoyance on the user's part by letting them know that their desire to have option 3 available on the website has been noted, and that it may be made available in the future. Perhaps you could even give a little text field for them to comment / complain, then the business can get valuable feedback in the user's own words (and if they are cursing them for providing a menu option to functionality that doesn't exist, then that might make them rethink things).

Delineate their options clearly

As others have noted, you need to separate this in the UI. You're only creating "friction" that will inhibit your conversion rate. You have two customers to deal with in this scenario:

Customer A: Just wants to place an order. Let them do it smoothly.
Customer B: Wants to communicate a preference. Let them understand what to expect when they chose this option.

Don't try to change behavior

Most online customers (the vast majority) don't want to talk to you. Let them contact you through the site. Offer the option to make their request via chat or email or an inline form. You can easily report back to MI with this data and it will probably be richer than the trick the business wants you to play on them.