I can virtually guarantee that "Big Engineering Company" (does it start with a W and end with an O?) has cultivated relationships with your customer's senior management. The big boys spend A LOT of money developing those relationships to achieve EXACTLY what you have described.
The big companies do not view quality the way you view it. To them, quality is simply a metric, an expense to be minimized. They are seeking to understand the absolute minimum quality standard that Customer will accept and they will shoot for that standard every time. Their only objective is to win the next job and book the revenue...that's it. Their sales teams are motivated by commissions and quarterly goals. You have a vested interest in the success of each and every project.
You are a small competitor that probably does good work with prices to match. You probably have no relationship, or maybe a very superficial relationship, with senior business unit leaders.
Who do you think will win big jobs, the supplier that invites Customer's senior managers to "Ultra Super Extreme Executive Management Celebration Galas at Fancy Hotels in Distant Cities" or a small SW vendor that does good cleanup work?
You have a marketing and networking problem. I recommend connecting with your customer's procurement division and work through them to meet business unit stakeholders. If they're incompetent procurement people they'll just hand you the names of key decision makers!
Context: I outsource services and negotiate commercial terms for a living. I handled global SW/HW engineering services for a number of years and dealt with MNCs and small local suppliers. It is possible to beat the big boys.