First things first, why do they want the carpet tiles taken up?
Commercial tiles cost between £25 - £65 per sqm to supply and fit, so this area has cost a lot of money to put down.
Have the tiles been cleaned lately? Perhaps they have been overwet? Maybe a change of use, so a vinyl tiled floor is now more practical.
How certain are you that there is very little adhesive to remove? If you base your costs on the edges, you may be in for a shock as you get further into the job!
Removing the glue will not be easy or quick, the chemicals required are very expensive, even more so if a large area is to be cleaned.
Then you have the very real risk of damage, some tiles are bound to have been stuck down rather firmly, then you run the risk of damage to the vinyl tiles underneath, if some break, they will not be easy to replace or match in colour, especially if the original vinyl has faded as well, you could then face a claim for replacing the entire floor.
If the carpet has been down a while, some of the carpet backing may well have welded itself to the vinyl, painstaking scraping would be the only way to remove this.
My advice on this one, would be to let the other company do the job, as £300 is nowhere near enough.
If you do go ahead with this, please go in with your eyes wide open.
If possible, give them a day rate for doing the job, put in a disclaimer to cover damage or unforseen circumstances and problems.
Dont forget if you use adhesive removing chemicals, that the vapurs from these chemicals are extremely dangerous, so ventilate the area very well, PPE must be worn at all times, these types of chemical are normally only used in small quantities.
Also consider the very real risk of exploision, these vapours are highly flammable, so allow lots and lots of time for the fumes to disperse before using machinery, mobiles, lights etc.
Without seeing the job myself, our charges based on what you have written would be between £1000 and £1500 + Vat.
Regards,
Rob