601.2f The player determines the total cost of the spell. Usually this is just the mana cost. Some spells have additional or alternative costs.
601.2g If the total cost includes a mana payment, the player then has a chance to activate mana abilities (see rule 605, “Mana Abilities”). Mana abilities must be activated before costs are paid.
That means you have to activate Phyrexian Altar before paying the costs, and removing the counters is part of the costs. So what you propose won't work.