Faster handling cannot supply missing credentials or resolve an inaccurate job description. Before spending more, identify whether the delay is evidence collection, legal review, an available government process, or travel preparation. Those are different bottlenecks with different owners.
Pay for the bottleneck you actually have
Request written quotes for document retrieval, translation, evaluation, and legal work, including what any rush charge changes. If an employer petition is appropriate, verify current expedited service rules and fees through USCIS with counsel; an expedited service is not an approval guarantee. Ask HR whether a revised start would avoid unnecessary expenses.
Keep the cost of changing travel or overlapping housing separate from the application budget. Identify the bottleneck before spending, because each type has a different owner and only some respond to money. Evidence collection responds to effort and sometimes to expedited retrieval; legal review responds to scope and availability; a government process responds to whatever rules apply to it; travel preparation responds to booking.
Paying to accelerate the wrong one buys nothing at all. Request written quotes for document retrieval, translation, evaluation and legal work, stating what any rush charge actually changes. Where an employer petition is appropriate, verify current expedited service rules and fees with counsel, remembering that an expedited service is not an approval guarantee.
Ask HR whether a revised start date would avoid the expense altogether, and keep travel and housing costs outside the application budget.