And the other problem is that Britain and France were not the only players. Russia was an as important player as Britain and in fact did much more to bring Napoleon's downfall than Britain did. And the point is that from 1803 to 1807, czar Alexander I wanted war at all costs and was in fact the soul of the coalition.
You're right that Russia is the key to a 1806/1807 peace. Fox's 1806 negotiations collapsed because Talleyrand demanded that Russia not be included in the Anglo-French discussions - a ploy to separate the coalition and play them off against eachother. In June, Britain found out that Russia was separately negotiating, and the French were increasing their demands with Britain (perhaps to try to pressure them back to the table).
One way of perhaps reconciling the dilemma here is playing Britain and Prussia against eachother - Prussia turned against France in 1806 because Napoleon told Talleyrand that he should be open to giving Hanover to Britain in exchange for peace. Prussia, who was 'temporarily' occupying Hanover at the time (officially "until the conclusion of a general peace"), would supposedly have been compensated for withdrawing from Hanover. The court of Prussia, which wanted to annex Hanover (perhaps in exchange for giving France the far western territories of Prussia), was greatly offended on learning of the plan, and despite Napoleon offering Prussia a sphere of influence in Northern Germany (a North German Confederation), this dispute over Hanover broke Prussia's trust in France, and was the casus belli for Prussia joining the Fourth Coalition.
Perhaps a sufficiently competent ATL French foreign minister could offer Hanover to Britain, keeping Britain interested and open to negotiations, and then frame Prussia's objections as breaking the terms of the agreement that allowed them to occupy Hanover. This would damage British-Prussian relations, and Britain might then join Russia in making peace at Tilsit (if Napoleon can be convinced to hold back on Prussia, as such a harsh treaty as OTL would scare Britain off and achieve the same result as OTL).
Another option would be France accepting Britain's demands for joint negotiations with Russia - OTL the June 1806 peace deal between France and Russia was vetoed by Alexander I, but if Napoleon and Alexander meet in person, Napoleon may be able to charm him into accepting a deal (as happened at Tilsit in OTL). The problem is that this puts France in a vulnerable position during negotiations - Russia was willing to cede Naples to France when negotiating alone, but together, Britain and Russia might pressure France to withdraw.
Of course, the devil is in the details for these peace treaties. OTL during the 1806 negotiations, France was willing to offer Hanover to Britain, plus recognition of British possession of Malta and the Cape Colony. Other issues, such as Naples being returned to Sicily, were the main focus of negotiations. As Britain grew more willing to accept French possession of Naples, Talleyrand increased tensions by demanding Sicily be given to France in exchange for Dalmatia, Ragusa and Albania (the latter two not being in French possession). Talleyrand also broke British trust by promising that no territorial changes in Germany would take place, and then after the July treaty forming the Confederation of the Rhine, he claimed that he only meant *after* the peace treaty is signed. If Talleyrand handled these disputes better, I could see a preliminary peace agreement based on France keeping Naples, Britain keeping Malta and the Cape Colony, and Hanover being transferred back to Britain despite Prussian objections.
Could the POD be that Charles Talleyrand, (French Minister of Foreign Affairs from 1799 to 1807) doesn't resign after the Treaty of Tilsit in 1807, so Napoleon's invasions of Portugal and Russia are avoided due to a better foreign policy? From what I've heard, Talleyrand was a great statesman during that time, serving France in the Pre-Revolutionary, Revolutionary, Napoleonic and Bourbon Restoration eras.
Talleyrand was very skilled, but also corrupt (as seen during the XYZ affair), and he had his own priorities beyond the interests of the French government. His style worked sometimes, but during the 1806 negotiations with Britain he was confrontational and demanding, poisoning relations with the British Whigs, and alienating Fox, perhaps the only British politician openly sympathetic with France.
There are two explanations for Talleyrand's actions in 1806 - that he failed by overestimating how far he could push Britain, or that he wanted negotiations to fail. If he wanted to fail, then he would have been acting against the wishes of Napoleon, who had told Talleyrand that he should be open to giving Hanover to Britain in exchange for peace. Neither option reflects well on Talleyrand.
Looking at the possibility of Talleyrand not resigning after Tilsit: Talleyrand may be able to convince Napoleon to not punish Prussia as much at Tilsit (the reason he resigned OTL), perhaps by letting Prussia retain Altmark-Magdeburg and/or Hanover. Talleyrand might convince Napoleon not to invade Spain or Russia, and he might bring Britain back to the table and negotiate a peace later. But his record in OTL (failure of the 1806 negotiations, failure to moderate the Treaty of Tilsit) doesn't give me much hope he would achieve any of this, either due to lack of ability or lack of motive.