Off-spinner Botha removed Imrul Kayes and Mohammad Ashraful in quick succession to leave Bangladesh 101-3.
Kayes came close to falling lbw to Swann, but Tamim for a while was unstoppable.
Peterhansel, driving a Mitsubishi, completed the 370km run from Bamako to Kayes, Mali, in three hours, 30 minutes and seven seconds.
Imrul Kayes played a more measured innings alongside Tamim, and then we saw Junaid Siddique and Jahurul Islam rallying round as well.
The competitors face a 630km ride in the 14th stage on Friday, including a 529km timed special between Kayes and Tambacounda, Senegal.
Tamim's 60 and 70 from Kayes got the innings off to a fine start, as Bangladesh capitalised on ragged bowling by India.
Kayes, however, is convinced the aspirin will be of the celebratory kind.
Mohammad Ashraful (29) and Kayes then put on 68 for the second wicket as Bangladesh went on to record their highest-ever one-day total against a Test playing nation.
Two balls later and Imrul Kayes looped a drive at another disguised delivery into the off-side, where Steve Finn swooped to his right to pouch the dipping ball.
The visitors' fielding was no more impressive than their bowling Harbhajan Singh's dropping a simple catch off the bowling of Ashish Nehra when Kayes was on just four runs.
Only Tendulkar put up any resistance, but even he had his moments, having been dropped in the slips by Imrul Kayes long before he had got into his stride.
BBC: Tendulkar sets record as Bangladesh make India struggle
Following Alastair Cook's dismissal for 21, Pietersen came in to join Trott and made 45 before spooning a catch into the hands of Imrul Kayes off Shakib Al Hasan.
England's bowlers, as they had been for much of Friday, were run ragged until Steven Finn removed both Tamim and Kayes in the run-up to tea on the fourth day.
Bangladesh lost the toss and were quickly in trouble, Tamim Iqbal falling to Boyd Rankin for five and Trent Johnston adding the scalps of Imrul Kayes (5) and Jahurul Islam (3).
Then, after 12 weeks' of hammering and chiselling, Kayes will witness Hatton contest the fight of his life before watching him slip into the layabout routine which has become his custom.
Imrul Kayes again paid the price for his fondness for the hook shot against Finn and, as in the first innings, picked out Ajmal Shazad, who took the catch at deep square leg.
With the follow-on enforced, Tamim and Kayes survived one or two hairy moments to get through the first 14 overs leading up to lunch, reaching 69-0, and the fun continued in the afternoon.
So once the weight is made, Kayes will be keeping a watchful eye on his charge to ensure Hatton is in the ring 28 hours later "bursting full of energy" with the right food in his system.
Bangladesh were inserted shortly after tea and they soon lost opener Imrul Kayes (1), caught behind off the bowling of Zaheer Khan, and Shahriar Nafees (21), caught at gully off Ishant Sharma as they embarked on an unlikely run chase.
It was important for England not to let the partnership get out of hand, but Swann failed to get his hands to a low Kayes edge off Anderson with the batsman on 30, which just about carried on the full.
Although he only took one wicket in Bangladesh's first innings and has taken just one so far in the second, he has performed better than his figures suggest, displaying good control and producing a beauty on Monday to dismiss Imrul Kayes.
It got even more exciting when Tamim slog-swept two sixes in an over off Swann, and as he eyed his third Test century Kayes reached his first fifty at this level with a well-executed sweep shot for four off the expensive Swann.
Ahead of the Olympics, Gavin had seen the nutritionist famous for regulating Ricky Hatton's weight, Kerry Kayes, and admitted there had been real worries he might not reach the required 60kg mark long before the team left for the holding camp in Macau.
In the Gazipur district north of Dhaka, Garments Export Village Ltd. was surprised to find its name on Wal-Mart's list and feared it would hurt the company's reputation, said Emrul Kayes Sujan, manager of human resources and compliance for Amtranet Group, the factory owner.
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