First you should fully understand your corporate travel policy. For example in my company there is a global policy then we have some regional policies and then we have management discretions and budgets as well.
you should always speak with your manager first he/she might dealt this type of problem before and there could be solutions already in place.
so if your corp policy dictates that you must book it through a corporate TA and it can only be economy class only then your options are limited. Nevertheless you can speak with your corp TA for upgrade options too.
what I never suggest is to go ahead with assumptions or incorrect invoices etc as you could end up paying your own travel fully etc.
also if you are unhappy with your options and you have to travel every week long haul economy then perhaps the job is not that great so you could look around. Most of the time I do ask questions around corp travel during my interview to have a sense of feeling what this potential employer policy is.