The disadvantage(s) are:

These answers address the question assuming the large corporation is the largest shareholder in the investment round or at least 40% of the investment offering.

Public perception: You raised the issue in your question. How many of your potential customers would perceive this negatively? You might actually consider doing some basic market research here with existing or potential customers.

Deal-structure and negative signal concerns: Often these investments come with terms that can weaken their appetite for investing. Also, if they decline to invest further, this will often be perceived as a negative signal (i.e. management didn't live up to its promise/expectations).

Influence in your business: Regardless of percentage interest, board control, etc., there will expect to wield influence in your product or service offering, and potentially (and most damagingly) make it difficult for you to sell to companies who they perceive as a threat, even if such terms aren't part of the deal.

No value-add: Often, these deals end-up creating little to no value, and this is most frustrating to a company when you made specific concessions and have had to deal with concerns I've raised above only to find the only thing they were good for was their money.

A great way to ensure you get the most value out of a corporate investor is for you to make the deal competitive and possibly bring in another corporate investor, or better yet, bring in great value-added investors whose total investment value makes the corporation's investment a small part of the round.

Happy to talk through the specifics of your concerns and ways to manage or address them in a call.

I think you have to follow your gut here. If its telling you that this investor is going to be a more of a headache and will take up most of your time then don't take the investment. You want to make sure you are focused on building a great company.
I would also have a conversation about how the investor feels about you approaching their competitors as potential customers. Get it all out on the table before signing the dotted line.

Several potential disadvantages:
1. A commitment to use the corporate's resources (even if inadequate)
2. Lack of flexibility to deviate from the corporate's plans and roadmaps when the market changes
3. Difficulty in getting other types of investors in down the road.