Praise be to Allah.
It is haraam for a person to outbid his brother or undercut him, because of the reports narrated by al-Bukhaari (2139) and Muslim (1412) from ‘Abdullah ibn ‘Umar (may Allah be pleased with him), according to which the Messenger of Allah (blessings and peace of Allah be upon him) said: “None of you should outbid his brother.”
Muslim (1414) narrated from ‘Uqbah ibn ‘Aamir that the Messenger of Allah (blessings and peace of Allah be upon him) said: “The believer is the brother of another believer, and it is not permissible for a believer to outbid [or undercut] his brother or propose marriage when his brother has already proposed, unless the latter gives up the idea.”
Muslim (1408) narrated from Abu Hurayrah that the Prophet (blessings and peace of Allah be upon him) said: “A man should not propose marriage to a woman to whom his brother has already proposed and he should not outbid [or undercut] his brother.”
An-Nawawi (may Allah have mercy on him) said: With regard to undercutting one’s brother, an example of that is saying to the one who has agreed to buy something, when he still has the option of cancelling the deal: Cancel this deal and I will sell you something similar for less than this man’s price, or I will sell you something of better quality for the same price, and the like. This is haraam.
It is also haraam to outbid one’s brother, which means saying to the seller, when he still has the option of cancelling the deal: Cancel this deal, and I will give you a higher price than this, and the like.
Another haraam practice is when the owner of the item and the one who wants to buy it have agreed on a price, but have not yet concluded the deal, someone else says to the seller: I will buy it from you. This is haraam after the price has been agreed upon." (Sharh Muslim 10/158).
Specifying that this applies to the time when the two parties still have the option to cancel the deal is one of the two scholarly views.
But the more correct view is that this prohibition includes that period and other times. No one has the right to outbid his brother, or to offer more rent than his brother, after the deal has been concluded, because this will lead to regret and will create grudges, and may even make someone look for a trick to cancel the deal. See: ash-Sharh al-Mumti‘ by Shaykh Ibn ‘Uthaymeen (8/204).
What you describe does not come under the heading of offering to purchase when someone else has already bought something, because the other person had not agreed to any deal with the owner.
And it does not come under the heading of buying after the deal has been concluded, because what is meant by that is that the two parties have agreed on a price, then someone comes and says: I will take it for a higher price, but that did not happen.
In fact, because this purchaser dragged his heels and did not want to buy for all this time, he is to be regarded as one who is no longer interested. The seller is not obliged to wait for him, and there is nothing to prevent anyone else from making an offer.
There is nothing wrong with your having bought the store, and you are not obliged to wait until the end of the month, as someone else suggested.
It should be noted that if the seller refused to sell to the tenant, or he rejected the price that he offered, there is no blame on anyone else for purchasing the store or making an offer, and the seller has no obligation towards the tenant.
Rather what is not allowed is to sell or make an offer when the seller has come to an agreement with the purchaser, or has accepted his offer and is happy with it, and they are still both content with the deal, as mentioned above.
As you have bought the shop, what this person is doing of trying to buy it is haraam and comes under the heading of outbidding his brother.
And Allah knows best.