Tenant lease expired; wants 6 day exten doesn't agree on terms FL

[J White]

This property is located in Florida.

We (Landlords) have a tenant that has been renting since 8/2011 signing 1 or 2 year lease renewals.  The last 2 year least started 8/1/17 and ended 7/31/19 with monthly rent $1700/year 1, $1650/year 2. However, at the start of May we notified the tenant that we intended to sell the property when the lease is up (7/31/19).  Tenant contacted us in the middle of July and said he/she was having difficulty finding another place to live so tenant asked for a 1 month lease extension.  We agreed and amended the 2 year lease agreement to end on 8/31/19 (signed by both landlord & tenant) & tenant paid the August 2019 rent via the standard procedure (deposit into our bank account).  Tenant was supposed to close on another property on 8/31/19 but due to the hurricane that closing was delayed so tenant asked to be able to stay in the property until 9/6/19 (which is the updated closing date).  This was relayed to us via our realtor (who will list the property once it is vacated).  We then sent tenant a written offer (via email) to stay in the property for $350 covering 9/1-9/6 requesting that it is payable to us by 8/31/19.  Tenant wants to stay and agreed to the amount but did not agree to the other terms insisting that "I’m not able to agree to depositing any funds into your account when you can clearly deduct this from the deposit you are and have been holding since 2011."(i.e. the security deposit).  We (landlords) do not and never did agree to this when talking with our realtor earlier who also spoke to the realtor that is assisting tenant with her purchase.  Tenant & her realtor are insisting on that there was agreement on this offer (i.e. money to be deducted from the security deposit) but that simply isn't the case.  So I have a few questions:

*Is there a valid agreement in place for tenant to remain in property & have this rent deducted from the security deposit?  Would a court rule in favor of the tenant (in terms of either her suing us for filing a security deposit claim > $350 or in the eviction process)?  My understanding is no, there is no agreement in place for her to remain in the property since we never agreed to those terms and never offered those terms

*If the answer to the above question is No, then at this point, tenant is considered a holdover tenant which per Florida statute 83.58 (http://www.leg.state.fl.us/statutes/index.cfm?App_mode=Display_Statute&URL=0000-0099/0083/Sections/0083.58.html) would allow us to collect double the amount of rent due for the time period that the tenant refuses to surrender possession of the property.  Is that correct?

*If the answers to above questions are "No, there is no valid agreement" and "Yes, this tenant is considered a holdover tenant entitling you to double the amount of rent", then what would be the rent amount used to determine the double amount?  Would it be the original $350 requested even though no official written agreement has been executed?  Or would it revert to the standard monthly rent of $1650/month since the 6 day lease extension agreement was never executed and thus, is not valid?  In other words, if that agreement never existed and was never offered, then it should clearly be the latter (i.e. it reverts to the terms of the most recent lease), but what happens in this situation where an amended lease is offered but not fully accepted?

For what it's worth, I'm not envisioning the tenant staying in the property long term or needing to evict tenant from the property (since that will take 6-7 weeks).  I'm just trying to get an idea in terms of what can be kept from the security deposit ($350? $1650?) after tenant does vacate the property.  And also if it is worth it to start the eviction process (i.e. file the 3 Day Notice) just as a safeguard should things go south down the road.

Thanks